Divisibility is one of the most fundamental concepts in number theory. A precise definition of what it means for a number to be divisible by another number is essential for defining other number-theoretic concepts such as that of prime numbers. In this handout, and the accompanying worksheet, you’ll find the basic definitions and key results about divisibility, primes and composite numbers, examples that illustrate these definitions, and problems to practice proof-writing skills by establishing various properties using nothing more than the definition of divisibility. Proofs of this type tend to be particularly simple, much like the proofs involving properties of even/odd numbers that you may have seen earlier in the course.

1. Divisibility

2. DON’T FORGET!
Be Punctual
Be Dressed/Groomed
Find a Distraction-free place
Be Respectful and Positive
Stay on Mute – Raise Hand if you want to Speak
Keep your Video ON
Stay Focused and on task

3. OBJECTIVES
At the end of this topic, as a pre-service teacher, you are
expected to:
A. discuss the definition and properties on divisibility;
B. define theorems related to divisibility; and
C. Solve problems of divisibility.
3

4. Definition. If a and b are integers with 𝒂 ≠ 𝟎,
we say that a divides b if there is an integer
c such that 𝒃 = 𝒂𝒄. If a divides b, we also
say that a is a divisor or factor of b and that
b is a multiple of a.
If a divides b, we write 𝒂|𝒃, while if a does NOT
divide b, we write 𝒂 ∤ 𝒃.

5. EXAMPLES
1. 𝟏𝟑 | 𝟏𝟖𝟐 (13 divides 182 or 13 is a divisor of 182)
2. −𝟐 | 𝟐𝟓𝟎
3. 𝟒 | 𝟏𝟎𝟎
4. 𝟔 ∤ 𝟒𝟒 (6 does not divide 44 or 6 is not a divisor of 44)
5. The divisors of 𝟏𝟐 are
± 𝟏, ±𝟐, ±𝟑, ±𝟒, ±𝟔, 𝒂𝒏𝒅 ± 𝟏𝟐
6. The divisors of 100 are
± 𝟏, ±𝟐, ±𝟒, , ±𝟓, ±𝟏𝟎, ±𝟐𝟎, ±𝟐𝟓, ±𝟓𝟎, 𝒂𝒏𝒅 ±
𝟏𝟎𝟎.
5

6. Theorem. If a, b, and c are integers with 𝒂|𝒃 and 𝒃|𝒄, then 𝒂|𝒄.
Proof. Because 𝒂|𝒃 and 𝒃|𝒄, there are integers e and f such that 𝒂𝒆 = 𝒃 and
𝒃𝒇 = 𝒄. Hence, 𝒄 = 𝒃𝒇 = 𝒂𝒆 𝒇 = 𝒂(𝒆𝒇), and we conclude that 𝒂|𝒄.
Example. Because 𝟏𝟏|𝟔𝟔 and 𝟔𝟔|𝟏𝟗𝟖, the previous theorem tells us that
𝟏𝟏|𝟏𝟗𝟖.
Theorem. If a, b, m, and n are integers, and if 𝒄|𝒂 and 𝒄|𝒃, then 𝒄|(𝒎𝒂 + 𝒏𝒃).
Proof. Because 𝒄|𝒂 and 𝒄|𝒃, there are integers e and f such that 𝒂 = 𝒄𝒆 and 𝒃 =
𝒄𝒇. Hence, 𝒎𝒂 + 𝒏𝒃 = 𝒎𝒄𝒆 + 𝒏𝒄𝒇 = 𝒄(𝒎𝒆 + 𝒏𝒇), Consequently, we see
that 𝒄|(𝒎𝒂 + 𝒏𝒃).
Example. As 𝟑|𝟐𝟏 and 𝟑|𝟑𝟑, previous theorem tells us that 3 divides 𝟓 ∗ 𝟐𝟏– 𝟑 ∗
𝟑𝟑 = 𝟏𝟎𝟓 − 𝟗𝟗 = 𝟔
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7. The Division Algorithm
> Theorem. If a and b are integers such that 𝒃 >
𝟎, then there are unique integers q and r such
that 𝒂 = 𝒃𝒒 + 𝒓 with 𝟎 ≤ 𝒓 < 𝒃.
> In this theorem, q is the quotient and r the
remainder, a the dividend and b the divisor. We
note that a is divisible by b if and only if the
remainder in the division algorithm is 0
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8. EXAMPLES
1. If 𝒂 = 𝟏𝟑𝟑 and 𝒃 = 𝟐𝟏, then 𝒒 = 𝟔 and 𝒓 = 𝟕, because
𝟏𝟑𝟑 = 𝟐𝟏 ∗ 𝟔 + 𝟕. We rewrite this as:
> 133
21
= 6 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
7
21
2. If 𝒂 = 𝟏𝟓𝟓 and 𝒃 = 𝟏𝟒, then 𝒒 = 𝟏𝟏 and 𝒓 = 𝟏,
because 𝟏𝟓𝟓 = 𝟏𝟒 ∗ 𝟏𝟏 + 𝟏. We rewrite this as:
> 155
14
= 11 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
1
14
8

9. EXAMPLES
3. If 𝒂 = 𝟏𝟒𝟕 and 𝒃 = 𝟏𝟔, then 𝒒 = 𝟗 and 𝒓 = 𝟑, because
𝟏𝟒𝟕 = 𝟏𝟔 ∗ 𝟗 + 𝟑. We rewrite this as:
> 147
16
= 9 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
3
21
9

10. The following examples display the quotient and remainder of a division.
1. Let 𝑎 = 1028 and 𝑏 = 34. Then 𝑎 = 𝑏𝑞 + 𝑟 with 0 ≤ 𝑟 <b, where
𝑞 =
1028
34
= 30 and
𝑟 = 1028 −
1028
34
∗ 34 = 1028 − 30 ∗ 34 = 8
Check: 1028 = 34 30 + 8
2. Let 𝑎 = −380 and 𝑏 = 75. Then 𝑎 = 𝑏𝑞 + 𝑟 with 0 ≤ 𝑟 <b, where
𝑞 =
−380
75
= −6 and
𝑟 = −380 −
−380
75
∗ 75 = −380 − −6 75 = 70.
Check: −380 = 75 −6 + 70
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11. The following examples display the quotient and remainder of a division.
3. Let 𝑎 = −3562 and 𝑏 = −232. Then 𝑎 = 𝑏𝑞 + 𝑟 with 0 ≤ 𝑟 <b,
where 𝑞 =
−3562
−232
= 15 and
𝑟 = −3562 −
−3562
−232
∗ −232 = −3562 − 15 −232 = −82.
Check: −3562 = −232 15 − 82
4. Let 𝑎 = 36724 and 𝑏 = −3647. Then 𝑎 = 𝑏𝑞 + 𝑟 with 0 ≤ 𝑟 <b,
where 𝑞 =
36724
−3647
= −11 and
𝑟 = 36724 − −11 −3647 = −3393.
Check: 36724 = −3647 −11 − 3393
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12. “The real fundamental
building blocks of the
universe and matter are
not indivisible particles,
but infinitely divisible
moments in time.”
― Khalid Masood
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13. 13
Thanks!
End of Presentation
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