ASVAB Study Guide
Basic Math · Lesson 5 of 6

Prime Numbers and Divisibility

Key rules and formulas, worked examples and 9 practice problems with step-by-step solutions.

Prime Numbers

Definition & Identification

p > 1, positive factors of p: {1, p}

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Example

Identify if 17 and 21 are prime numbers.

Show solution

1) 17 is only divisible by 1 and 17 -> Prime.

2) 21 is divisible by 1, 3, 7, 21 -> Composite (Not prime).

Practice problems

  1. Problem 1

    Which of the following is a prime number?

    Show solution

    Answer: 13

    13 has only two factors (1 and 13). 15 is divisible by 3 and 5.

  2. Problem 2

    What is the smallest prime number?

    Show solution

    Answer: 2

    2 is the smallest prime number. 1 is neither prime nor composite.

  3. Problem 3

    How many prime numbers are there between 10 and 20?

    Show solution

    Answer: 4

    The prime numbers between 10 and 20 are 11, 13, 17, 19 (total of 4).

Remember

2 is the only EVEN prime number (and the smallest prime). 0 and 1 are NOT prime numbers!

Divisibility Rules for 2 and 5

Last digit

Last digit ∈ {0, 2, 4, 6, 8} (for 2)

Last digit ∈ {0, 5} (for 5)

A number is divisible by 2 if its last digit is even. It is divisible by 5 if its last digit is 0 or 5.

Example

Check divisibility of 240 by 2 and 5.

Show solution

Last digit is 0 -> 240 is divisible by both 2 and 5.

Practice problems

  1. Problem 4

    Which number is divisible by 5?

    Show solution

    Answer: 145

    145 ends in 5, so it is divisible by 5.

  2. Problem 5

    Is 3,578 divisible by 2?

    Show solution

    Answer: Yes, because it ends in 8

    A number is divisible by 2 if its last digit is even (8 is even).

  3. Problem 6

    Find the value of x if $$4{,}56x$$ is divisible by both 2 and 5.

    Show solution

    Answer: 0

    To be divisible by both 2 and 5, the last digit x must be 0.

Remember

If a number ends in 0, it is divisible by BOTH 2 and 5 (hence divisible by 10).

Divisibility Rules for 3 and 9

Sum of digits

Sum of digits ÷ 3 (for 3)

Sum of digits ÷ 9 (for 9)

Add all the digits together; if the sum is divisible by 3 (or 9), then the number itself is divisible by 3 (or 9).

Example

Check if 2,457 is divisible by 9.

Show solution

1) Sum of digits: $$2 + 4 + 5 + 7 = 18$$.

2) 18 is divisible by 9 -> 2,457 is divisible by 9.

Practice problems

  1. Problem 7

    Is 123 divisible by 3?

    Show solution

    Answer: Yes $$(1+2+3 = 6)$$

    Sum of digits = $$1 + 2 + 3 = 6$$. Since 6 is divisible by 3, 123 is divisible by 3.

  2. Problem 8

    Which number is divisible by 9?

    Show solution

    Answer: 459

    $$4+5+9 = 18$$ (divisible by 9). $$4+5+3 = 12$$ (divisible by 3, but not 9).

  3. Problem 9

    Find the digit x such that 3,x45 is divisible by 9.

    Show solution

    Answer: 6

    Sum of digits = $$3 + x + 4 + 5 = 12 + x$$. For $$12 + x$$ to be divisible by 9, $$x = 6 (12 + 6 = 18)$$

Remember

If a number is divisible by 9, it is ALWAYS divisible by 3. But a number divisible by 3 is NOT necessarily divisible by $$9!$$