Chapter Review: Drill Sets
Drill Set 1
1. Is 4,005 divisible by 5?
2. Does 51 have any factors besides 1 and itself?
3. x = 20
The prime factors of x are:
The factors of x are:
4. Is 123 divisible by 3?
5. Does 23 have any factors besides 1 and itself?
6. x = 100
The prime factors of x are:
The factors of x are:
7. Is 285,284,901 divisible by 10?
8. Is 539,105 prime?
9. x = 42
The prime factors of x are:
The factors of x are:
10. Is 9,108 divisible by 9 and/or by 2?
11. Is 937,184 prime?
12. x = 39
The prime factors of x are:
The factors of x are:
13. Is 43,360 divisible by 5 and/or by 3?
14. Is 81,063 prime?
15. x = 37
The prime factors of x are:
The factors of x are:
16. Determine which of the following numbers are prime numbers. Remember, you only need to find one factor other than the number itself and 1 to prove that the number is not prime.
Drill Set 2
1. If x is divisible by 33, what other numbers is x divisible by?
2. The prime factorization of a number is 3 × 3 × 7. What is the number and what are all its factors?
3. If x is divisible by 8 and by 3, is x also divisible by 12?
4. If 40 is a factor of x, what other numbers are factors of x?
5. The only prime factors of a number are 5 and 17. What is the number and what are all of its factors?
6. 5 and 6 are factors of n. Is n divisible by 15?
7. If 64 divides n, what other divisors does n have?
8. The prime factorization of a number is 2 × 2 × 3 × 11. What is the number and what are all its factors?
9. 14 and 3 divide n. Is 12 a factor of n?
10. If x is divisible by 4 and by 15, is x a multiple of 18?
11. 91 and 2 go into n. Does 26 divide n?
12. n is divisible by 5 and 12. Is n divisible by 24?
13. If n is a multiple of both 21 and 10, is 30 a divisor of n?
14. 4, 21, and 55 are factors of n. Does 154 divide n?
15. If n is divisible by 196 and by 15, is 210 a factor of n?
Drill Set 3
Simplify the following expressions by combining the terms.
1. x5 × x3
2. 76 × 79
3. 32 × 35
4. 92 × 94
5. 
6. 
7. 4−2 × 45
8. 
9. 
10. 75 × 53
Combine the following expressions.
11. x2 × x3 × x5
12. 34 × 32 × 3
13. y3 × y−5
14. 
15. 
16. y7 × y8 × y−6
17. 
18. 62 × 6−7 × 64
19. 
20. 
Simplify the following expressions by combining the terms.
21. (a3)2
22. (22)4
23. (32)−3
24. (52)x
25. (y3)−4
Combine the following expressions.
26. (x2)6 × x3
27. y3 × (y3)−4
28. 
29. (z6)x × z3x
30. 
Rewrite each negative exponent as a positive exponent.
31. x−2
32. 4−4
33. y−4z−4
34. 6−3
35. x5 × x−9
Drill Set 4
Combine the following expressions and solve for x.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
Simplify the following roots.
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
Drill Set 5
Simplify the following expressions.
1. 83 × 26
2. 34 × 95
3. 492 × 77
4. 43 × 85
5. 118 × 1212x
6. 254 × 1253
7. 9−2 × 272
8. 2−7 × 82
9. 73x × 49−3
10. 45x × 32−2x
11. x3 = 27
12. y2 = 81
13. 2x3 = 128
14. z2 + 18 = 54
15. 3x5 = 96
Solve the following equations.
16. 3x = 81
17. 6y−3 = 36
18. 74x−11 = 7
19. 5−4 = 252x
20. 42 = 163y−8
Drill Set Answers
Drill Set 1:
1. Is 4,005 divisible by 5?
Yes: 4,005 ends in 5, so it is divisible by 5.
2. Does 51 have any factors besides 1 and itself?
Yes: The digits of 51 add up to a multiple of 3 (5 + 1 = 6), so 3 is a factor of 51. Thus, 51 has factors besides 1 and itself.
3. x = 20
The prime factors of x are:
The factors of x are:
| Small | Large |
| 1 | 20 |
| 2 | 10 |
| 4 | 5 |
4. Is 123 divisible by 3?
Yes: The digits of 123 add up to a multiple of 3 (1 + 2 + 3 = 6), so 123 is divisible by 3.
5. Does 23 have any factors besides 1 and itself?
No: 23 is a prime number. It has no factors besides 1 and itself.
6. x = 100
The prime factors of x are:
The factors of x are:
| Small | Large |
| 1 | 100 |
| 2 | 50 |
| 4 | 25 |
| 5 | 20 |
| 10 | 10 |
7. Is 285,284,901 divisible by 10?
No: 285,284,901 ends in a 1, not a 0, so it is not divisible by 10.
8. Is 539,105 prime?
No: 539,105 ends in a 5, so 5 is a factor of 539,105. So are 1 and 539,105. Prime numbers have only two factors, so 539,105 is not prime.
9. x = 42
The prime factors of x are:
The factors of x are:
| Small | Large |
| 1 | 42 |
| 2 | 21 |
| 3 | 14 |
| 6 | 7 |
10. Is 9,108 divisible by 9 and/or by 2?
Yes and Yes: The digits of 9,108 add up to a multiple of 9 (9 + 1 + 0 + 8 = 18), so it is divisible by 9. 9,108 ends in 8, so it is even, which means it is divisible by 2.
11. Is 937,184 prime?
No: 937,184 ends in 4, which means it's even. Therefore, it's divisible by 2. It's also divisible by 1 and by 937,184. Prime numbers have only two factors, so 937,184 is not prime.
12. x = 39
The prime factors of x are:
The factors of x are:
| Small | Large |
| 1 | 39 |
| 3 | 13 |
13. Is 43,360 divisible by 5 and/or by 3?
Yes and No: Because 43,360 ends in 0, it is divisible by 5. The digits of 43,360 do not add up to a multiple of 3 (4 + 3 + 3 + 6 + 0 = 16), so it is not divisible by 3.
14. Is 81,063 prime?
No: The digits of 81,063 add up to a multiple of 3 (8 + 1 + 0 + 6 + 3 = 18), so 3 is a factor of 81,063, however, 1 and 81,063 are also factors of 81,063. Prime numbers have only two factors, so 81,063 is not prime.
15. x = 37
The prime factor of x is 37.
The factors of x are (remember that 1 is not a prime number!):
| Small | Large |
| 1 | 37 |
16. The numbers in bold below are prime numbers:
Prime numbers: 2, 3, 5, 7, 17, 29, 31
Not prime:
All of the even numbers other than 2 (6, 10, 258, 786), since they are divisible by 2.
All of the multiples of 5 that haven't already been ruled out (15, 655, 1325).
All of the remaining numbers whose digits add up to a multiple of 3, since they are divisible by 3, by definition: 9 (digits add to 9), 21 (digits add to 3), 27 (digits add to 9), 33 (digits add to 6), 303 (digits add to 6), 1,023 (digits add to 6). Again, all six numbers are divisible by 3.
Drill Set 2:
1. If x is divisible by 33, what other numbers is x divisible by?
If x is divisible by 33, then x is also divisible by everything 33 is divisible by. The factors of 33 are:
| Small | Large |
| 1 | 33 |
| 3 | 11 |
So x is also divisible by 1, 3, and 11.
2. The prime factorization of a number is 3 × 3 × 7. What is the number and what are all its factors?
3 × 3 × 7 = 63, which means the number is 63.
| Small | Large |
| 1 | 63 |
| 3 | 21 |
| 7 | 9 |
3. If x is divisible by 8 and by 3, is x also divisible by 12?
Yes: For x to be divisible by 12, it must contain the same prime factors as 12, which are 2 × 2 × 3. Therefore, 12 contains two 2’s and a 3. x also contains two 2’s and a 3, therefore x is divisible by 12.
4. If 40 is a factor of x, what other numbers are factors of x?
If 40 is a factor of x, then any factor of 40 is also a factor of x.
| Small | Large |
| 1 | 40 |
| 2 | 20 |
| 4 | 10 |
| 5 | 8 |
5. The only prime factors of a number are 5 and 17. What is the number and what are all its factors?
If 5 and 17 are the only prime factors of the number, then the number equals 5 × 17, which means the number is 85.
| Small | Large |
| 1 | 85 |
| 5 | 17 |
6. 5 and 6 are factors of n. Is n divisible by 15?
Yes: For n to be divisible by 15, you need to know that it contains the same prime factors as 15. 15 = 3 × 5. Therefore, 15 contains a 3 and a 5. Because n also contains a 3 and a 5, n is divisible by 15.
7. If 64 divides n, what other divisors does n have?
If 64 divides n, then any divisors of 64 will also be divisors of n.
| Small | Large |
| 1 | 64 |
| 2 | 32 |
| 4 | 16 |
| 8 | 8 |
8. The prime factorization of a number is 2 × 2 × 3 × 11. What is the number and what are all its factors?
2 × 2 × 3 × 11 = 132
| Small | Large |
| 1 | 132 |
| 2 | 66 |
| 3 | 44 |
| 4 | 33 |
| 6 | 22 |
| 11 | 12 |
9. 14 and 3 divide n. Is 12 a factor of n?
Cannot Tell: For 12 to be a factor of n, n must contain all the same prime factors as 12. 12 = 2 × 2 × 3, so 12 contains two 2’s and a 3. Although n also contains a 3, it only contains one 2 that you know of, so you don't know whether 12 is a factor of n.
10. If x is divisible by 4 and by 15, is x a multiple of 18?
Cannot Tell: For x to be a multiple of 18, x would have to be divisible by 18. For x to be divisible by 18, it has to contain all the same prime factors as 18: 18 = 2 × 2 × 3 × 3, so 18 contains two 2’s and two 3’s. Although x contains two 2’s, it only contains one 3 that you know of, so you don't know whether x is a multiple of 18.
11. 91 and 2 go into n. Does 26 divide n?
Yes: For 26 to divide n, n has to contain all the same prime factors as 26: 26 = 2 × 13, so 26 contains a 2 and a 13. Because n also contains a 2 and a 13, 26 divides n.
12. n is divisible by 5 and 12. Is n divisible by 24?
Cannot Tell: For n to be divisible by 24, it has to contain all the same prime factors as 24: 24 = 2 × 2 × 2 × 3, so 24 contains three 2’s and a 3. Although n contains a 3, it only contains two 2’s that you know of, so you don't know whether n is divisible by 24.
13. If n is a multiple of both 21 and 10, is 30 a divisor of n?
Yes: For 30 to be a divisior of n, n has to contain all the same prime factors that 30 contains: 30 = 2 × 3 × 5, so 30 contains a 2, a 3, and a 5. Because n also contains a 2, a 3, and a 5, 30 is a divisor of n.
14. 4, 21, and 55 are factors of n. Does 154 divide n?
Yes: For 154 to divide n, n has to contain all the same prime factors as 154: 154 = 2 × 7 × 11, so 154 contains a 2, a 7, and an 11. Because n also contains a 2, a 7, and an 11, 154 divides n.
15. If n is divisible by 196 and by 15, is 210 a factor of n?
Yes: For 210 to be a factor of n, n must contain all the same prime factors as 210: 210 = 2 × 3 × 5 × 7, so 210 contains a 2, a 3, a 5, and a 7. n contains a 2, a 3, a 5, and a 7, so 210 is a factor of n.
Drill Set 3
1. x8: x5 × x3 = x (5+3) = x8
2. 715: 76 × 79 = 7 (6+9) = 715
3. 37: 32 × 35 = 3 (2+5) = 37
4. 96: 92 × 94 = 9 (2+4) = 96
5. 52: 
= 5 (5−3) = 52
6. 5−2: 
= 5(3−5) = 5−2
7. 43: 4−2 × 45 = 4(−2+5) = 43
8. (−3)(a − 2): 
= (−3)(a − 2)
9. 11(4 − x): 
= 11(4 − x)
10. Can't simplify: 75 × 53 = no common bases or exponents!
11. x10: x2 × x3 × x5 = x(2 + 3 + 5) = x10
12. 37: 34 × 32 × 3 = 3(4 + 2 + 1) = 37
13. y−2: y3 × y−5 = y(3 − 5) = y−2
14. x9: 
= x9
15. 54 x + 2: 
16. y9: y7 × y8 × y−6 = y(7 + 8 + (−6)) = y9
17. x7: 
18. 6−1: 62 × 6−7 × 64 = 6(2 + (−7) + 4) = 6−1
19. z10: 
20. 38x + 3y: 
21. a6: (a3)2 = a (3 × 2) = a6
22. 28: (22)4 = 2(2 × 4) = 28
23. 3−6: (32)−3 = 3 (2 × −3) = 3−6
24. 52x: (52)x = 5 (2 × x) = 52x
25. y−12: (y3)−4 = y (3 × −4) = y−12
26. x15: (x2)6 × x3 = x(2 × 6 + 3) = x(12 + 3) = x15
27. y−9: y3 × (y3)−4 = y(3 + 3 × −4) = y(3 + (−12)) = y−9
28. 36: 
29. z9x: (z6)x × z3x = z(6 × x + 3x) = z(6x + 3x) = z9x
30. 5y + 3 : 
31. 
32. 
33. 
34. 
35. 
Drill Set 4
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
Drill Set 5
1. 215: 83 × 26 = (23)3 × 26 = 29 × 26 = 215
2. 314: 34 × 95 = 34 × (32)5 = 34 × 310 = 314
3. 711: 492 × 77 = (72)2 × 77 = 74 × 77 = 711
4. 221: 43 × 85 = (22)3 × (23)5 = 26 × 215 = 221
5. 114x + 8: 118 × 1212x = 118 × (112)2x = 118 × 114x = 114x + 8
6. 517: 254 × 1253 = (52)4 × (53)3 = 58 × 59 = 517
7. 32: 9−2 × 272 = (32)−2 × (33)2 = 3−4 × 36 = 32
8. 2−1: 2−7 × 82 = 2−7 × (23)2 = 2−7 × 26 = 2−1
9. 73x − 6: 73x × 49−3 = 73x × (72)−3 = 73x × 7−6 = 73x − 6
10. 1: 45x × 32−2x = (22)5x × (25)−2x = 210x × 2−10x = 20 = 1
11. 3: x3 = 27
x = 3
12. 9 or −9: y2 = 81
y = 9 or −9
13. 4: 2x3 = 128
x3 = 64
x = 4
14. 6 or −6: z2 + 18 = 54
z2 = 36
z = 6 OR −6
15. 2: 3x5 = 96
x5 = 32
x = 2
16. 4: 3x = 81
3x = 34
x = 4
17. 5: 6y − 3 = 36
6y − 3 = 62
y − 3 = 2
y = 5
18. 3: 74x − 11 = 71
4x − 11 = 1
4x = 12
x = 3
19. −1: 5−4 = 252x
5−4 = (52)2x
5−4 = 54x
−4 = 4x
−1 = x
20. 3: 42 = 163y − 8
42 = (42)3y − 8
42 = 46y − 16
2 = 6y − 16
18 = 6y
3 = y
OR
(22)2 = (24)3y − 8
24 = 212y − 32
4 = 12y − 32
36 = 12y
3 = y
OR
42 = 163y − 8
16 = 163y − 8
161 = 163y − 8
1 = 3y − 8
9 = 3y
3 = y