Modular Arithmetic
Introduction to Modular
Arithmetic
The congruent symbol consisting of three horizontal bars was introduced in print in 1801 by Carl
Friedrich Gauss (1777–1855).
Modulo 𝒏
Two integers 𝑎 and 𝑏 are said to be congruent modulo 𝒏, where 𝑛 is a natural number, if
𝑎−𝑏
𝑛
is an integer. In this case, we write 𝑎 ≡ 𝑏 mod 𝑛. The number 𝑛 is called modulus. The
statement 𝑎 ≡ 𝑏 mod 𝑛 is called a congruence.
Example: Determine whether a congruence is true.
Determine whether the congruence is true.
a. 29 ≡ 8 mod 3
b. 15 ≡ 4 mod 6
Solution:
a.
𝑎−𝑏
𝑛
=
29−8
3
=
21
3
=7
Since 7 is an integer, then 29 ≡ 8 mod 3 is a true congruence.
b.
𝑎−𝑏
𝑛
=
15−4
6
Since
=
11
6
11
6
is not an integer, then 15 ≡ 4 mod 6 is not a true congruence.
Example: A Day of the Week
July 4, 2017, was a Tuesday. What day of the week is July 4, 2022?
Solution:
There are 5 years between the two dates. Each year has 365 days except 2020, which has one extra
day because it is a leap year. So, the total number of days between the two 5 ∙ 365 + 1 = 1826.
Because 1826 ÷ 7 = 260 remainder 6, 1826 ≡ 6 mod 7. Any multiple of 7 days past a given day
will be the same day of the week. So, the day of the week 1826 days after July 4, 2017, will be the
same as the day 6 days after July 4, 2017. Thus, July 4, 2022, will be a Monday.
2022 – 2017 = 5 𝑦𝑒𝑎𝑟𝑠
5 ∙ 365 + 1 = 1826
1826 ÷ 7 = 260
With remainder 6
1826 ≡ 6 mod 7
July 4, 2022, will be a Monday.
Carpio, Joezerk A.
PEC 1A
PED 113
Objective:
At the end of a 60-minute discussion, the Grade 10 – Einstein students are expected to
do the following with 85% level of performance:
a. Evaluate whether a congruence is true.
b. Calculate a day of the week.
Audience
Grade 10 – Einstein students
Behavior
evaluate, calculate
Condition
At the end of a 60-minute discussion
Degree of Performance
85% level of performance