Solutions

1. Yes

   If a is divisible by 7 and by 18, its prime factors include 2, 3, 3, and 7, as indicated by the factor tree to the right. Therefore, any integer that can be constructed as a product of any of these prime factors is also a factor of a. Thus, 42 = 2 × 3 × 7. Therefore, 42 is also a factor of a.

   

2. Cannot Be Determined

   If r is divisible by 80, its prime factors include 2, 2, 2, 2, and 5, as indicated by the factor tree below. Therefore, any integer that can be constructed as a product of any of these prime factors is also a factor of r. Thus, 15 = 3 × 5. Because the prime factor 3 is not in the factor tree, you cannot determine whether 15 is a factor of r. As numerical examples, you could take r = 80, in which case 15 is not a factor of r, or r = 240, in which case 15 is a factor of r.

   

3. Yes

   If two numbers are both multiples of the same number, then their sum is also a multiple of that same number. Because n and p share the common factor 7, the sum of n and p must also be divisible by 7.

4. Cannot Be Determined

   For 8 to be a factor of 2g, you would need two more 2’s in the factor tree. By the Factor Foundation Rule, g would need to be divisible by 4. You know that g is not divisible by 8, but there are certainly integers that are divisible by 4 and not by 8, such as 4, 12, 20, 28, and so on. However, while you cannot conclude that g is not divisible by 4, you cannot be certain that g is divisible by 4, either. As numerical examples, you could take g = 5, in which case 8 is not a factor of 2g, or g = 4, in which case 8 is a factor of 2g.

   

5. Cannot Be Determined

   If j is divisible by 12 and by 10, its prime factors include 2, 2, 3, and 5, as indicated by the factor tree to the right. There are only two 2’s that are definitely in the prime factorization of j, because the 2 in the prime factorization of 10 may be redundant—that is, it may be the same 2 as one of the 2’s in the prime factorization of 12.

   

   Thus, 24 = 2 × 2 × 2 × 3. There are only two 2’s in the prime box of j; 24 requires three 2’s. Therefore, 24 is not necessarily a factor of j.

   As another way to prove that you cannot determine whether 24 is a factor of j, consider 60. The number 60 is divisible by both 12 and 10. However, it is not divisible by 24. Therefore, j could equal 60, in which case it is not divisible by 24. Alternatively, j could equal 120, in which case it is divisible by 24.

6. Cannot Be Determined

   If xyz is divisible by 12, its prime factors include 2, 2, and 3, as indicated by the factor tree to the right. Those prime factors could all be factors of x and y, in which case 12 is a factor of xy. For example, this is the case when x = 20, y = 3, and z = 7. However, x and y could be prime or otherwise not divisible by 2, 2, and 3, in which case xy is not divisible by 12. For example, this is the case when x = 5, y = 11, and z = 24.

   

7. Yes

   By the Factor Foundation Rule, if 6 is a factor of r and r is a factor of s, then 6 is a factor of s.

8. Yes

   By the Factor Foundation Rule, all the factors of both h and k must be factors of the product, hk. Therefore, the factors of hk include 2, 2, 2, 2, 2, 3, and 7, as shown in the combined factor tree to the right. Thus, 21 = 3 × 7. Both 3 and 7 are in the tree. Therefore, 21 is a factor of hk.

   

9. Yes

   The fact that d is not divisible by 6 is irrelevant in this case. Because 12 is divisible by 6, 12d is also divisible by 6.

   

10. Cannot Be Determined

    If u is divisible by 60, its prime factors include 2, 2, 3, and 5, as indicated by the factor tree to the right. Therefore, any integer that can be constructed as a product of any of these prime factors is also a factor of u. Next, 18 = 2 × 3 × 3. There is only one 3 in the factor tree, therefore, you cannot determine whether or not 18 is a factor of u. As numerical examples, you could take u = 60, in which case 18 is not a factor of u, or u = 180, in which case 18 is a factor of u.

    

11. (C)

    The prime factorization of 40 is 2 × 2 × 2 × 5. So 40 has two distinct prime factors: 2 and 5. The prime factorization of 50 is 5 × 5 × 2, so 50 also has two distinct prime factors: 2 and 5. Therefore, the two quantities are equal.

12. (B)

    Simplify Quantity A first. There is only one even prime number: 2. Therefore, Quantity A is 12 × 2 = 24.

    Quantity A	Quantity B
    The product of 12 and an even prime number =	The sum of the greatest four factors of 12 =
    12 × 2 = 24	12 + 6 + 4 + 3 = 25

    The four greatest factors of 12 are 12, 6, 4 and 3. Thus, 12 + 6 + 4 + 3 = 25. Therefore, Quantity B is greater.

13. (C)

    When 20 is divided by 12, the result is a quotient of 1 and a remainder of 8 (12 × 1 + 8 = 20).

    When 32 is divided by 12, the result is a quotient of 2 and a remainder of 8 (12 × 2 + 8 = 32).

    x = 20, y = 32, and z = 12

    Quantity A	Quantity B
    8	8

    Therefore, the two quantities are equal.

14. 9

    Because  has a remainder of 4, b must be at least 5 (remember, the remainder must always be smaller than the divisor). The smallest possible value for a is 4 (it could also be 9, 14, 19, etc.). Thus, the smallest possible value for a + b is 9.

15. 0

    Because has a remainder of 0, x is divisible by y. Therefore, xz will be divisible by y, and so will have a remainder of 0 when divided by y.