Check Your Skills Answer Key

  1. No

    Is 123,456,789 divisible by 2?

    123,456,789 is an odd number, because it ends in 9, so 123,456,789 is not divisible by 2.

  2. Yes

    Is 732 divisible by 3?

    The digits of 732 add up to a multiple of 3 (7 + 3 + 2 = 12), so 732 is divisible by 3.

  3. No

    Is 989 divisible by 9?

    The digits of 989 do not add up to a multiple of 9 (9 + 8 + 9 = 26), so 989 is not divisible by 9.

  4. No

    Every whole hundred is divisible by 4, so you only need to check the amount “left over.” Because 78 is not divisible by 4, then 4,578 is not divisible by 4.

  5. Yes

    Any number divisible by both 2 and 3 is divisible by 6. So 4,578 must be divisible by 2, because it ends in an even number. It also must be divisible by 3, because the sum of its digits is a multiple of 3 (4 + 5 + 7 + 8 = 24). Therefore, 4,578 is divisible by 6.

  6. Yes

    Easiest to use your calculator for this one: 603,864 ÷ 8 = 75,483 with no remainder.

    Alternatively, evaluate the three-digit number at the end; every whole thousand is divisible by 8, so you only need to check the amount “left over.” 864 is divisible by 8 because 864 = 800 + 64 and both 800 and 64 are multiples of 8.

  1. Find all the factors of 90.

    Small Large
    1 90
    2 45
    3 30
    5 18
    6 15
    9 10

  2. Find all the factors of 72.

    Small Large
    1 72
    2 36
    3 24
    4 18
    6 12
    8 9

  3. Find all the factors of 105.

    Small Large
    1 105
    3 35
    5 21
    7 15

  4. Find all the factors of 120.

    Small Large
    1 120
    2 60
    3 40
    4 30
    5 24
    6 20
    8 15
    10 12

  1. List all the prime numbers between 20 and 50.

    23, 29, 31, 37, 41, 43, and 47

  1. Find the prime factorization of 90.

  2. Find the prime factorization of 72.

  3. Find the prime factorization of 105.

  4. Find the prime factorization of 120.

  1. The prime factorization of a number is 3 × 5. What is the number and what are all its factors?

    3 × 5 = 15

    Small Large
    1 15
    3 5

  2. The prime factorization of a number is 2 × 5 × 7. What is the number and what are all its factors?

    2 × 5 × 7 = 70

    Small Large
    1 1 70 2 × 5 × 7
    2 2 35 5 × 7
    5 5 14 2 × 7
    7 7 10 2 × 5

  3. The prime factorization of a number is 2 × 3 × 13. What is the number and what are all its factors?

    2 × 3 × 13 = 78

    Small Large
    1 1 78 2 × 3 × 13
    2 2 39 3 × 13
    3 3 26 2 × 13
    2 × 3 6 13 13

For questions 19−21, x is divisible by 24.

  1. Must Be True

    x is divisible by 6

    For x to be divisible by 6, you need to know that it contains the same prime factors as 6, which contains a 2 and a 3. x also contains a 2 and a 3, therefore, x must therefore be divisible by 6.

  2. Could Be True

    x is divisible by 9

    For x to be divisible by 9, you need to know that it contains the same prime factors as 9, which contains two 3’s. However, x only contains one 3 that you know of. But the question mark means x may have other prime factors, and may contain another 3. For this reason, x could be divisible by 9.

  3. Must Be True

    x is divisible by 8

    For x to be divisible by 8, you need to know that it contains the same prime factors as 8, which contains three 2’s. Because x also contains three 2’s, x must therefore be divisible by 8.

For questions 22−24, x is divisible by 28 and by 15.

  1. Must Be True

    x is divisible by 14.

    For x to be divisible by 14, you need to know that it contains the same prime factors as 14, which contains a 2 and a 7. Because x also contains a 2 and a 7, x must therefore be divisible by 14.

  2. Must Be True

    x is divisible by 20.

    For x to be divisible by 20, you need to know that it contains the same prime factors as 20, which contains two 2’s and one 5. Because x also contains two 2’s and a 5, x must therefore be divisible by 20.

  3. Could Be True

    x is divisible by 24.

    For x to be divisible by 24, you need to know that it contains the same prime factors as 24, which contains three 2’s and one 3. However x contains one 3, but only two 2’s that you know of. But the question mark means x may have other prime factors, and may contain another 2. For this reason, x could be divisible by 24.

  1. 1

    The number 6 goes into 13 two full times, which means the quotient is 2. Therefore, 2 × 6 = 12, and 12 + 1 = 13. The remainder is 1.

  2. 14

    For a number to result in a remainder of 4 when divided by 5, it has to be equal to a multiple of 5, plus 4. The first of these is 4 (5 × 0 + 4 = 4), the second is 9 (5 × 1 + 4 = 9), and the third is 14 (5 × 2 + 4). Thus, 14 is the first double-digit number that produces the required remainder.

  3. 7

    Using the Remainder Formula:

    Therefore, x + y = 9 (Q + Q′) + 7 and the remainder is 7.

  4. 3

    Again using the Remainder Formula:

    Therefore, xy = (9Q + 4)(9Q′ + 3) = 81QQ′ + 27Q + 36Q′ + 12.

    Because each of the terms except 12 is divisible by 9, and a 9 can be removed from 12, the correct answer is 12 – 9 = 3.