Student
Explorations
TEACHER VERSION • Modular Arithmetic
in Mathematics
Modular
Arithmetic
This issue of Student Explorations in
Mathematics will introduce students to
modular arithmetic and its applications.
The lesson provides opportunities to apply
and extend ideas to unfamiliar contexts. Students will begin with familiar problems involving
an analog clock and will soon realize that they already
have some experience with modular arithmetic. Using this
knowledge, mathematical notation and ideas are developed
to help students solve problems involving basic number theory. Applications to serial number
coding are also discussed.
TEACHER
NOTES
✍“Modular Arithmetic” offers opportunities for students to make progress
toward the Standards for Mathematical Practices outlined in the Common
Core State Standards for Mathematics. However, any one activity, including this one, will not necessarily result in students achieving the depth of a
Standard. Students must have multiple opportunities to engage in a variety
of lessons before they can achieve or be proficient in a particular Standard.
✍ Suggested answers are in red. Instructional notes are in blue and
are preceded by the hand icon. These two features do not appear in the
Student Version.
✍ For questions denoted with a
, it might be helpful to allow students
to first make sense of the questions for themselves and to then share their
responses with a partner or in a small group. At your discretion, you can
then follow up with a whole-group discussion.
✍ Supplemental materials include the following files:
Student Version.pdf
Add Integers.gsp
Modular Arithmetic.xls
Student
Explorations
TEACHER VERSION • Modular Arithmetic
in Mathematics
Modular Arithmetic
Introduction
What is the sum of 8 and 7? Do you know that in some contexts
it can be 3? How do you figure out when you will meet your friend if
the time is 8:00 now and you will meet her in 7 hours?
In this activity, we will learn the mathematics behind problems such as this and how to apply the mathematical understanding to other situations.
Elementary Arithmetic
1. What is the sum of 5 and 2? What is the sum of 7 and 5? What is the sum of 8 and 26?
7, 12, and 34
✍ These familiar problems are designed to serve as an entry point and help students experience the difference
between elementary arithmetic and modular arithmetic.
2. Show how you could arrive at your answers using the number lines below.
✍ This is a good opportunity to highlight to students the connections between the symbolic representation (2 + 5 = 7 )
and a visual (or pictorial) representation of the problem. Ask students why arrowheads are drawn on each end of the
number line.
5
+2
-3 -2 -1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
7
+5
-3 -2 -1
0
1
2
8
+ 26
-3 -2 -1
0
1
2
We would like to thank Dina Zolotusky for the ideas she contributed to this SEM activity.
Copyright © 2012 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only.
1
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
3. Let’s add time using a typical 12-hour clock. Show how
you arrived at your answers using the figures below.
a. It is now 5 o’clock; what time will it be in 2 hours?
7 o’clock
b. What time will it be 5 hours after 7 o’clock?
12 o’clock
4. Compare your answers from questions 1, 2, and 3.
Explain why the sum of 5 and 2 (or 7 and 5) is the same
on the number line and on the clock, but the sum of 8
and 26 is not.
Responses will vary. Sample responses may include the
following:
In questions 1 and 2, the number lines extend forever, so
no matter what two numbers are added, their sum can
be found on the number line. The number line contains
all numbers that are part of simple arithmetic questions,
and it also contains all necessary answers. For question 3,
the clock numbers include only numbers 1–12. So, on the
clock, the numbers of the arithmetic question and answer
must be only from 1–12. (We can allow 26 in the clock
problem by regrouping it as 12 + 12 + 2.) So, when summing 5 and 2 (or 7 and 5), all the numbers are found on
a 12-hour clock. But for 8 + 26, the result on the number
line (34) is not found on the clock, so the answer is not the
same as on the number line.
5. In the space at the bottom of this page, create a number line for question 3c, and then show your work on it.
Compare the number line you invented with
those in your group. Identify the similarities and
differences between the number lines.
See the suggested number line below. Because the
12‑hour clock has only numbers 1–12, these numbers
must “start over” after 12.
c. What is 26 hours after 8 o’clock?
10 o’clock
✍ Ask students to compare and contrast the “structure”
of the number line in questions 2 and 5. Below are
additional questions to consider:
• Why is the zero missing in the response to question 5?
• Why is the left arrowhead on the number line in
question 5 missing?
At this juncture, students should have identified the
difference between elementary and clock arithmetic:
Specifically, the set of numbers in clock arithmetic is
5. Number line for a 12 hour clock
1 2 3 4 5 6 7 8 9 10 1112 1 2 3 4 5 6 7 8 9 10 1112 1 2 3 4 5 6 7 8 9 10 1112 1 2 3 4
2
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
limited to the whole numbers 1–12. If students have
not made this connection, work toward this understanding before proceeding.
The clocks from question 3 should be familiar. Recall that
one full day on Earth is 24 hours, which is represented
by twice around the 12-hour clock. For questions 6–15,
consider the following scenario:
Time is quite different on the planet Orlandus. One full
day on Orlandus is 30 hours, which is twice around a
15-hour clock.
6. Explain which of the following clocks would not be
useful to tell time on Orlandus.
Responses will vary. Sample responses may include the
following.
Share your reasons with your group.
a. A 12-hour clock
Clock a would not be useful because it does not cover a
full day (or even half a day) on Orlandus.
7. In a way, the clock is a type of counting machine. We
can count up or count down. Using a 15-hour clock,
what are the results when you—
a. subtract 5 from 11?
6
b. add 15 to 5?
5
c. add 47 to 7?
9
✍ As needed, encourage students to re-create a
15-hour clock to aid in visualizing the calculations.
hink about how you would answer question 7 using a
T
15‑hour clock or a 30-hour clock. Mathematicians solve
big problems by understanding little ones. One strategy
you can use when a problem seems overwhelming is to
adapt the strategy that worked when solving a simpler
and more familiar problem. Let’s do that here to develop
a strategy. Think about a 12-month calendar.
8. Izzy’s family planned a vacation 25 months from
July 2012. What year and month is their family vacation?
Explain your strategy.
b. A 15-hour clock
Clock b would be uselful because two complete
revolutions would be the same as one day, like our
current clock.
25 months after July 2012 is August 2014.
✍ M any students may continue to “count up”
25 months (method 1): “August, then September,
then October, then November, then….” Help these
students consider other counting strategies.
✍ M ost students will probably choose to count up by
c. A 30-hour clock
Clock c would be useful because it represents one full day
on Orlandus.
d. A 50-hour clock
Clock d would not be useful because it covers more than
one day on Orlandus.
✍ Some students may suggest clock c. You may want
to introduce students to the 24-hour clock used in
many industries worldwide. They may question why
the 12-hour clock is generally used around the world
since a 24-hour clock better represents one full
day of time. At your discretion, share the historical
significance of how the 12-hour clock was adopted
or the geometry associated with assigning 24 hours
on a circle.
3
groups of 12 months (method 2). That is, starting
from July 2012, counting 12 + 12 + 1 months yields
August 2014. Question 9 is designed to help students
experience cognitive dissonance with this process
and prompt them to seek an alternate approach.
Some students may be ready to report 25 = 2(12) + 1
(method 3) or 25  12 = 2 r l (method 4). Assess their
understanding by asking for an explanation of their
work. Ask students to consider how the strategy of
working on a familiar 12-month calendar (or a 12-hour
clock) helps to understand adding and subtracting on
a 3-hour or a 30-hour clock.
9. Izzy’s mom purchased a savings bond that is fully
mature after 1000 months. Name the year and month
that is 1000 months after July 2012.
Explain your strategy to a group member.
1000 months after July 2012 is November 2095.
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
✍ 1 000 ÷ 12 = 83(12) r 4
83(12) + 4 = 1000
83 years and 4 months
This is a good opportunity to elicit student responses
and discuss student strategies as a whole class.
Students should experience that method 1 (counting
up) is inefficient. Help students to progress toward
method 2 (grouping).
10. Apply your strategy to answer the following questions.
a. Today is Monday; what day will it be in 13 days?
13 = (7) + 6. One week and six days after Monday is
Sunday.
b. It is now December; what month was it 26 months
ago?
26 = (12 + 12) + 2. Two years and two months before
December is October.
c. It is 8 o’clock now; what time will it be in 100 hours?
100 = (12 + 12 + 12 + 12 + 12 + 12 + 12 + 12) + 4.
Four hours after 8 o’clock is 12 o’clock.
d. Izzy had 67 sillybands and gave 8 of his friends the
same number of sillybands. How many were left?
67 = (8 + 8 + 8 + 8 + 8 + 8 + 8 + 8) + 3. Izzy will have
3 leftover bands.
✍ As a means of differentiating your instruction, you
may choose to provide organizers (clock diagrams,
calendars, manipulatives, etc.) to help your students
visualize their approach to this problem. Monitor
students’ work, ensuring that they are progressing
toward method 2 (or beyond). Help students understand the connection between 67 = 8(8) + 3 (method 3) and 67 ÷ 8 = 8 r 3 (method 4). Some students
may approach question 10.d as a typical long division
problem: 67 ÷ 8 = 8 r 3 (method 4).
11. Izzy decides to share his 53 M&M’s® with four friends.
This situation can be explained mathematically by
53 = (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5) + 3.
Based on the situation, explain what—
a. the values of 3 and 53 represent.
The 3 represents the leftover M&M’s once everyone gets
an equal amount, and 53 represents the total number of
M&M’s that Izzy had before he shared them.
b. the value of 5 represents.
The 5 represents the total number of people to whom the
M&M’s were distributed.
✍ Alternatively, 5 can mean the total number of M&M’s
distribute all the M&M’s. You may choose to have
students physically do this activity and reference it
during classroom discussions.
c. t he value that (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5)
represents.
This shows that 50 M&M’s were evenly divided among
5 people.
✍ Additionally, it explains that it took 10 turns of giving
each of his 5 friends an M&M before he was left with
just 3.
Notice that the situation in question 11 can be more
compactly written as 53 = 10(5) + 3. In either case,
understanding what each value {3, 5, 10, and 53}
represents is important.
12.Try this: Write a story (or a situation) that can be
explained by 115 = 7(15) + 10.
Responses will vary. Sample responses may include
the following: If the teacher distributes 115 pencils to
15 students, then each student will get 7 pencils, and
the teacher will have 10 left.
13. How is the situation in question 12 the same as a
division problem? Try it: Divide 115 by 15.
115 divided by 15 is 7 with a remainder of 10.
7 r10
15 115
)
14. Explain how the value of the remainder in question 13
is related to the 10 in question 12.
The remainder in the division problem is the same as the
“leftover” number in your story.
Notice that 115 = 7(15) + 10 can also be written as
p = q(d ) + r. This is known as the integer division algorithm, and it shows how the product, p, is related to the
divisor, d, quotient, q, and remainder, r.
15. Rewrite your work in question 11 in p = q(d ) + r form:
53 = 10 (5) + 3
✍ You may need to help some students understand that
p = q(d ) + r is another way of writing this:
q+r
d p
)
Let’s go back to question 10 and rewrite the solutions
using the division algorithm form. For example, If today is
Monday, what day will it be in 13 days? The solution for
this example is 13 = 1(7) + 6.
given out during each of the 10 turns it took to
4
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
• If it is now December, what month was it 26 months
ago?
26 = 2(12) + 2
• If it is 8 o’clock now, what time will it be in 100 hours?
100 = 8(12) + 4
• Izzy had 67 sillybands and gave 8 of his friends the
same number of sillybands. How many were left?
67 = 8(8) + 3
Responses may vary. Sample answers may include that
based on this development, the symbol “mod” means
identify the remainder when you divide the product (44)
by the divisor (12).
✍ You may want to share that “mod” is an invented
shorthand for a set of arithmetic operations. It is
akin to inventing a shorthand notation to describe
(2 + 3) = 5 and 5 × 2 × 3 = 30 as (p + q)(p)(q) = p@q.
Here, the invented shorthand @ means multiply the
sum by each addend. In fact, some programming
languages use the symbol % to represent the mod
operation.
When different problems can be solved using the same
strategy, mathematicians try to understand the similarities among the problems and develop a rule. This rule
can then be used to answer future questions that have
the same characteristics or similarities. We have one
representation of that rule as p = q(d) + r. Let’s examine
another representation.
18. The mod symbol is itself shorthand for the Latin word
modulus, which means a small measure. Explain why
you think mathematicians may have chosen modulus
or mod to describe the relationship between 44, 12,
and 8 in question 17.
Making a Generalization or Rule
✍ The shorthand mod describes the small amount (or
16. Identify the key information needed to answer
question 10. How is this information related to the
general terms you used in question 15?
In each case, you had to know the amount (13, 26, 100,
and 67). This is product p. In each case, you needed to
know the value after which the numbers started to “cycle
over” (7 days, 12 months, 12 hours, and 8 friends). This
is the divisor d. In each case, you needed to know the
remainder (6 days after Monday is Sunday, 2 months
before December is October, etc.) This is the remainder r.
✍ This understanding will help prompt the need for the
“mod” notation that follows.
Because the remainder r is dependent on the product p
and divisor d, mathematicians invented a way to write this
using a shortcut or symbol:
p(mod d ) = r , or p ≡ r (mod d )
This is read as “p mod d is r.” The question, “What hour
is it 44 hours after 2 o’clock on a 12-hour clock?” is
answered by 44(mod 12) = 8. That is, the solution is the
same as 8 hours past 2 o’clock, or 10 o’clock.
17.What arithmetic operations does the shorthand “mod”
represent? For instance, the symbol 3 in the expression 6 3 3 means multiply 6 by 3 (thus, add 6 to itself
3 times).
Responses may vary.
remainder) that is left when you divide the product
(44) by the divisor (12). Note that the meaning of mod
varies slightly based on the mathematical context. For
instance, in number theory, mod is not the remainder
but the divisor. In computer programming, mod is
simply the remainder. In clock or modular arithmetic,
mod represents the number after which the integers
“wrap around” after they reach a certain value.
Alternatively, Drexel’s Math Forum states mod as
the “amount by which a number exceeds the largest
integer multiple of the divisor that is not greater than
that number.”
Modular Arithmetic
When we perform mod operations, we are doing modular
arithmetic. Sometimes it is referred to as clock arithmetic
(although it can be applied to more than just clocks)
because numbers “wrap around” when they reach a fixed
quantity.
19. Let’s do some modular arithmetic. Evaluate the
following:
a. 10(mod 3) =
10(mod 3) = 1 since 3(3) + 1 = 10
b. 55(mod 7) =
55(mod 7) = 6 since 7(7) + 6 = 55
c. 27(mod 9) =
27(mod 9) = 0 since 9(3) + 0 = 27
Share your idea from question 17 with your group.
5
d. 96(mod 13) =
96(mod 13) = 5 since 7(13) + 5 = 96
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
✍ Students may object to question 19d., stating that
having “negative clock values,” or “negative items to
divide,” is impossible. Help students to move beyond
the context used to develop the mod operation.
20. Determine the values of d so that 15(mod d ) = 3.
d = {4, 6, 12}
✍ Students may complete this in a rote manner. Urge
them to develop a strategy.
21. a. Izzy passes out 60 pieces of candy. Determine the
value of d so that 60(mod d ) = 0.
If 60(mod d ) has a value of 0, then the remainder is 0
when 60 is divided by d, which occurs when d is a
factor of 60. Thus, we need to find all factors of 60:
d = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}.
b. For what value of p is p(mod 21) always 0?
If p(mod 21) = 0, then the remainder of p ÷ 21 must be 0.
This occurs when p is a multiple of 21. There are an infinite number of possibilities, of which some are p = {21,
42, 63,…}. Or simply for all nonzero integers n, p = {21n}.
c. Determine all possible values of d ≤ 4 so that
2 ≤ 60 (mod d) ≤ 4.
We are looking for when the value of 60(mod d ) is equal to
2, 3, or 4. That is, for what values of d does 60 ≤ d result
in a remainder of 2, 3, or 4? This occurs when d = {7, 8,
14, 19, 28, 29, 56, 57, 58}.
Applications of Modular Arithmetic
Did you know that mod, modulo, or modulus is used often
in real life? Modular arithmetic is often used in coding
information. The United States Postal Service (USPS) uses
modular arithmetic to detect errors or forgeries of money
orders. You can view a copy of a money order online at
https://www.usps.com/shop/accepting-money-orders.htm.
The serial number of a USPS money order is 11 digits
long. The first 10 digits of the serial number (8810024532)
are followed by a security check digit (7). If the money
order is genuine, then the 10-digit number modulo 9 will
be equal to the check digit.
22. Money order A has a serial number of 51177875501.
Money order B has a serial number of 88100245327.
Using modular arithmetic, determine if the money
orders are genuine.
For Money order A, the serial number is genuine if the first
10 digits of the number is 10-digit code(mod 9) = 1.
Notice that 5117787550 = 568643061(9) + 1. Therefore,
5117787550(mod 9) = 1. Thus, the serial number corresponds with a genuine money order.
6
For Money order B, notice that 8810024532 =
978891614(9) + 6. Therefore, 8810024532(mod 9) ≠ 7.
Thus, the serial number does not correspond with a
genuine money order.
✍ You may want to ask students why the official
US Post Office website is displaying an image of a
money order whose serial number is ingenuous.
23. R
eferring back to question 22, what is the next largest
serial number that can be issued for Money order A
using the modulo 9 security check system with the
same check digit?
Describe your solution strategy to your group.
The serial number must be 11 digits long with the last
digit being 1. So 10-digit number(mod 9) = 1 or simply
p(mod 9) = 1. We also know from above that 5117787550
= 568643061(9) + 1 produces a genuine serial number.
One solution method is to realize that the product (serial
number) changes when the quotient (568643061) changes.
The quotient is the number of multiples of 9 that divide into
the product and produce a remainder of 1. Therefore,
we can increase the quotient by 1 and compute:
p = 568643062(9) + 1 = 5117787559. Thus, the next
largest serial number is 51177875591.
✍ Encourage students who are struggling to use the
problem-solving strategy from questions 7 and 8.
Ask what it means to have a remainder of 1. How
are remainders created? If we can identify all values
of p that yield a reminder of 0, does this help us find
values of p that produce remainders of 1?
24. H
ow can we produce a list of all the possible serial
numbers for USPS money orders with the check digit
of 1? Assume that the serial number needs to begin
with a nonzero digit.
There are 111,111,111 serial numbers that are 11 digits
long and pass the security check. The smallest serial
number is 10000000001, and the largest is 99999999911.
✍ This is a good problem to help students persevere
in problem solving using the strategies and skills
developed in previous questions. One strategy that
problem solvers use when a rule is sought to describe a seemingly infinite number of possibilities
is to start somewhere—anywhere! We cannot solve
the entire problem at once, but we can solve smaller
problems and build from there.
For this problem, money order serial numbers have
a check digit of 1, so we know that p(mod 9) = 1.
Before we consider this, let’s revisit the simpler
scenario of p(mod 9) = 0.
The reminder is 0 when p is a multiple of 9. That is,
p = q(9) + 0, where q is any positive integer. We
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
must develop serial numbers for a check digit of 1,
so we can modify our initial statement to p = q(9) + 1
(for all positive integers q).
The first few p values that follow the “multiple of 9
plus 1” rule are
0(9) + 1 = 1,
1(9) + 1 = 10,
2(9) + 1 = 19,
3(9) + 1 = 28.
Although these pass the 10(mod 9) = 19(mod 9) =
28(mod 9) = 1 test, the numbers are not 10 digits
long. Our work, however, does help us write a rule:
10-digit number = (some multiple)(9) + 1
Letting n take the place of “some multiple,” the
expression n(9) + 1 helps produce the 10-digit
numbers. What are the smallest and largest values
of n so that n(9) + 1 is 10 digits long? The smallest
and largest 10-digit values are 1,000,000,000 and
9,999,999,999. Thus, 1,000,000,000 < n(9) + 1 <
9,999,999,999. Solving the compound inequality, we
find that
111,111,111 ≤ n ≤
The car number coding for this receipt is coded as
mod(0192050, 7) = 5.
✍ We need to identify the check digit, determine the
modulus, and then perform the test. The car number
is 01920505 with the last digit (5) assigned as the
check digit. We know that 192050(mod d ) = 5. One
method for solving this is trial and error. This strategy is relatively fast when the trials are small in number. Because the modulus is a single-digit integer
[1, 9], the statement 192050(mod d ) = 5 is false for
all values except 7. Thus, the modulus is 7. Finally,
192050(mod 7) = 5, so the receipt is genuine.
Another method is to determine the modulus analytically using skills and strategies from prior problems.
Given 192050(mod d) = 5, we also know that (d + 5)
must be a factor of 192050. If 1 ≤ d ≤ 9, then 1 + 5 ≤
d + 5 ≤ 9 + 5 or 6 ≤ d + 5 ≤ 14. Omitting all values
greater than 9 yields d = {6, 7, 8, 9}. But d must be
unique. Only four values are feasible, so we can
apply trial and error to identify the correct modulus.
Doing so reveals 7 as the modulus.
9,999,999,998
.
9
Because n must be an integer, we can apply the
greatest integer function (i.e., the floor function) and
conclude that
 9, 999, 999, 998 

 = 1,111,111,110.
9
Thus, n is an integer such that 111,111,111 ≤ n ≤
1,111,111,110. There are 999,999,999 possible
values of n that produce a 10-digit number, but only
111,111,111 (dividing by 9) pass the p(mod 9) = 1
test. The smallest and largest 10-digit numbers are
(111,111,111)(9) + 1 and (1,111,111,110)(9) + 1.
All (valid) serial numbers can be viewed as the
10-digit number produced by n(9) + 1 with an
additional 1 placed after the rightmost digit to
display the check digit.
Thus, the serial numbers are
{10000000001,
10000000091,
10000000181,
………………99999999911}.
25. Many other companies also include a security check
to guard against forgeries. Avis® uses a single-digit
modulo with a single-digit check number as the car
identification number. Identify the check digit for this
car rental and determine the modulus if the receipt is
genuine.
7
Can you …
• investigate
how modular arithmetic is used to create
ISBNs (International Standard Book Numbers) for books?
• complete the addition and multiplication tables on the
next page using mod 6? The values of (1 + 3)(mod 6) and
(4 + 5)(mod 6) are provided.
Student Explorations in Mathematics, May 2012
TEACHER VERSION • Modular Arithmetic
+
0
1
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5
0

0
1
2
3
4
5
0
1
4
1
2
2
3
3
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5
• explain the difference between modular arithmetic and
number-base arithmetic?
• create
unique 15-digit serial numbers (with a 2-digit
check code) to encode a class set of calculators?
• research
the method known as casting out nines?
Did you know that …
• modular arithmetic was introduced by mathematician
Carl Friedrich Gauss in 1801?
•G
auss developed a formula that uses modular arithmetic
to predict the date of Easter Sunday?
•m
odular arithmetic can be used to prove the divisibility
rules for 3 and 9?
• international bank account numbers (IBANs) use modulo
97 to identify account errors?
Deaf Education Access for Computational Science.
2002. “Modular Mathematics.” The Shodor Education
Foundation. http://www.shodor.org/succeedhi/succeedhi/
modularmath/introduction-content.html.
“A Latin Dictionary.” Perseus Digital Library, Tufts University.
Oxford: Clarendon Press, 1879. http://www.perseus.tufts
.edu/hopper/text?doc=Perseus%3Atext%3A1999.04
.0059%3Aentry%3Dmodulus.
MathBoys. “Math Words, and Some Other Words of
Interest: Definition of Modulus.” http://pballew.net/
arithme1.html.
Mathematica Ludibunda. “Smart Joe.”
http://mathematica.ludibunda.ch/smart-joe6.html.
United States Postal Service. 2012. “Accepting Money
Orders.” https://www.usps.com/shop/accepting-moneyorders.htm.
Mathematical Content
Resources
Ask Dr. Math. “Mod, Modulo, and Modular Arithmetic.”
Drexel University Goodwin College of Professional
Studies. http://mathforum.org/library/drmath/view
/62930.html.
Elementary arithmetic; Clock arithmetic; Modular arithmetic;
Integer division algorithm; Factors; Basic number theory;
Check digit; Real-world applications; Problem solving.
Student Explorations in Mathematics is published electronically by the National Council of Teachers of Mathematics, 1906 Association Drive, Reston,
VA 20191-1502. The five issues per year appear in September, November, January, March, and May. Pages may be reproduced for classroom use
without permission.
Editorial Panel Chair:
Co-Editor: Editorial Panel: Field Editor: Board Liaison:
Editorial Manager:
Production Editor: Production Specialist: 8
Mary Lou Metz, Indiana University of Pennsylvania, mlmetz@iup.edu
Cheryl Adeyemi, Virginia State University, cadeyemi@vsu.edu
Darshan Jain, Adlai E. Stevenson High School, Illinois, djainm7712@gmail.com
Larry Linnen, University of Colorado–Denver, llinnen@q.com
Sharon McCready, Department of Education, Nova Scotia, Canada, mccreasa@gov.ns.ca
Anthony Stinson, Clayton State University, Morrow, GA, anthonystinson@clayton.edu
Ed Nolan, Montgomery County Public Schools, Rockville, Maryland, edward_c_nolan@mcpsmd.org
Latrenda Knighten, Melrose Elementary School, Baton Rouge, Louisiana, ldknighten@aol.com
Beth Skipper, NCTM, bskipper@nctm.org
Luanne Flom, NCTM
Rebecca Totten, NCTM
Student Explorations in Mathematics, May 2012