MA 118: Chapter 4 – Number Theory
Section 4.2
Reminder: If d divides or is a factor of a number n,
the number can be written n = d·k for some number
k.
Theorem: Divisibility of Sums and Differences
If d, a, and b are natural numbers and d divides both
a and b then d divides both their sum a  b and
difference and a b .
Theorem: Divisibility of a Product
Let a and b be natural numbers with no common
factor other than 1 (Greatest Common Factor
GCF = 1). Then, if a and b are both divisors of c,
then ab divides c.
Theorem Divisibility by 2 and 5
If n is a natural number, then n is divisible by 2 if and
only if its units digit is 0, 2, 4, 6, or 8. If n is a
natural number, then n is divisible by 5 if and only if
its units digit is 0 or 5.
Section 4.2: Tests for Divisibility
Rule: Divisibility by 10
If n is a natural number, then n is divisible by 10 if
and only n is divisible by 2 and 5 , that is if and only
if its units digit is 0.
Rule: Divisibility by 4, and 8
A natural number n is divisible by 4 if and only if 4
divides the number named by the last two digits of n.
Similarly, 8 divides n, if and only if 8 divided the
number named by the last 3 digits of n. In general, 2r
divides n if and only if 2r divides the number named
by the last r digits of n.
Examples: Is n = 7,293,748 divisible by 2, 4, 8 and
16?
Rule: Divisibility by 3 and 9
A natural number n is divisible by 3 if and only if the
sum of its digits is divisible by 3. Similarly, a natural
number is divisible by 9 if and only if the sum of its
digits is divisible by 9.
Example: 37,284
Section 4.2: Tests for Divisibility Rule:
Divisibility by 6
If n is a natural number, then n is divisible by 6 if and
only n is divisible by 2 and 3.
Rule: Divisibility by 11
A natural number n is divisible by 11 if and only if
the difference of the sums of its digits in the even and
odd positions of the number is divisible by 11.
Examples: n = 7,293,748 and n = 121
Combined Test for Divisibility by 7, 11, and 13
Example: n = 92,252,191,213
Step 1 – Break up the number into 3 digit groups:
Group 1(odd)
Group 2(even) Group 3(odd)
Group 4 (even)
092
252
191
213
Step 2 – Add the even position digits and the odd
0 9 2
2 5 2
position digits:
191
 213
283
4 6 5
Step 3 – Find the difference between the sums:
4 6 5
 283
18 2
Step 4 – The original number n = 92,252,191,213 is
divisible by 7 if and only the difference 182 is
divisible by 7.
The original number n = 92,252,191,213 is
divisible by 11 if and only the difference 182 is
divisible by 11.
The original number n = 92,252,191,213 is
divisible by 13 if and only the difference 182 is
divisible by 13.
Example: n = 7,293,748
Section 4.3 Greatest Common Divisors (GCD)
and Least Common Multiples (LCM)
Cuisenaire rods for finding d  GCD(a,b) and
m  LCM(a,b) .
Definition: If a and b are natural numbers. The
largest natural number d that divides both a and b is
called their greatest common divisor or greatest
common factor, written d  GCD(a,b) or
d  GCF(a,b) .
Finding the greatest common divisor from
intersection of sets is useful when the numbers are
small.
Example: Find the greatest common divisor of 12
and 42. Find D12 {
} and D42 {
}.
Theorem: GCDs from Prime Power Representation
a a
Let a  p11 p22... prar be the prime power representation
of a and
b b
b  p11 p22... prbr
be the prime power
c c
GCD(a,b)  p11 p22... prcr
representation of b. Then
where ck is the lesser of ak and bk for 0  k  r .
Section 4.3 Greatest Common Divisors (GCD) and
Least Common Multiples (LCM)
Examples: Find GCD(12,42) .
Find GCD(350,420) .
If a is divided by b, we obtain a  bq  r, 0  r  b .
Solving for r, r  a bq . It turns out that a and b have
the same set of divisors as b and r.
Theorem The GCD and the Division Algorithm
Let a and b be any two natural numbers and let q and
r be determined by the division algorithm. So
a  bq  r, 0  r  b . Then GCD(a,b)  GCD(b,r) .
Example: Find GCD(124,2940) . Check using prime
power representation. Notice the last non-zero
remainder is the GCD.
This theorem gives the Euclidean algorithm theorem.
Section 4.3 Greatest Common Divisors (GCD) and
Least Common Multiples (LCM)
Theorem The Euclidean Algorithm
Let a and b be any two natural numbers. Using the
division algorithm, find q1, q2,. . . qs and r1, r2,. . . rs-1
so that
a  bq1  r1
0  r1  b
b  r1q2  r2
0  r2  r1
r1  r2q3  r3
0  r3  r2
.
.
rs3  rs2qs1  rs1
0  rs1  rs2
rs2  rs1qs  0
Then GCD(a,b)  rs1.
Section 4.3 Greatest Common Divisors (GCD) and
Least Common Multiples (LCM)
Definition: If a and b are natural numbers. The
smallest natural number m that is a multiple of both a
and b is called their least common multiple, written
m  LCM(a,b) .
Finding the least common multiple from intersection
of sets is useful when the numbers are small.
Example: Find the least common multiple of 12 and
42. Find M12 {
} and M 42 {
}.
Theorem: LCMs from Prime Power Representation
a a
Let a  p11 p22... prar be the prime power representation
b b
of a and b  p11 p22... prbr be the prime power
c
c
representation of b. Then LCM(a,b)  p11 p22... prcr
where ck is the larger of ak and bk for 0  k  r .
Example: Find LCM(12,42) .
Section 4.3 Greatest Common Divisors (GCD) and
Least Common Multiples (LCM)
Notice that
12 42  22 3 237  23 22 37  684  GCD(12,42) LCM(12,42)
Theorem: If a and b are natural numbers
ab  GCD(a,b)LCM(a,b) .
Example: Use the Euclidean algorithm and the fact
that LCM(a,b)  ab
to find LCM(124,2940) .
GCD(a,b)