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Math  /  Numbers & Operations

QuestionPRNGIC Write the prime factorization of the number. If the number is prime, write prime.
1. 42
2. 104
3. 75
4. 23
6. 27
7. 72 a. 180
9. 47
11. 88
12. 49
13. 83
14. 142

Studdy Solution

STEP 1

1. A prime factorization involves expressing a number as a product of its prime factors.
2. We will use methods like trial division to find the prime factors.
3. If a number is prime, we will simply state "prime."

STEP 2

1. Prime factorization of 42.
2. Prime factorization of 104.
3. Prime factorization of 75.
4. Determine if 23 is prime.
5. Prime factorization of 27.
6. Prime factorization of 72.
7. Prime factorization of 180.
8. Determine if 47 is prime.
9. Prime factorization of 88.
10. Prime factorization of 49.
11. Determine if 83 is prime.
12. Prime factorization of 142.

STEP 3

Prime factorization of 42: 42=2×2142 = 2 \times 21

STEP 4

Continue factorization of 21: 21=3×721 = 3 \times 7

STEP 5

Combine the factors: 42=2×3×742 = 2 \times 3 \times 7

STEP 6

Prime factorization of 104: 104=2×52104 = 2 \times 52

STEP 7

Continue factorization of 52: 52=2×2652 = 2 \times 26

STEP 8

Continue factorization of 26: 26=2×1326 = 2 \times 13

STEP 9

Combine the factors: 104=23×13104 = 2^3 \times 13

STEP 10

Prime factorization of 75: 75=3×2575 = 3 \times 25

STEP 11

Continue factorization of 25: 25=5×525 = 5 \times 5
STEP_{10}: Combine the factors: 75=3×5275 = 3 \times 5^2
STEP_{11}: Determine if 23 is prime. Since 23 has no divisors other than 1 and itself, it is prime.
STEP_{12}: Prime factorization of 27: 27=3×927 = 3 \times 9
STEP_{13}: Continue factorization of 9: 9=3×39 = 3 \times 3
STEP_{14}: Combine the factors: 27=3327 = 3^3
STEP_{15}: Prime factorization of 72: 72=2×3672 = 2 \times 36
STEP_{16}: Continue factorization of 36: 36=2×1836 = 2 \times 18
STEP_{17}: Continue factorization of 18: 18=2×918 = 2 \times 9
STEP_{18}: Continue factorization of 9: 9=3×39 = 3 \times 3
STEP_{19}: Combine the factors: 72=23×3272 = 2^3 \times 3^2
STEP_{20}: Prime factorization of 180: 180=2×90180 = 2 \times 90
STEP_{21}: Continue factorization of 90: 90=2×4590 = 2 \times 45
STEP_{22}: Continue factorization of 45: 45=3×1545 = 3 \times 15
STEP_{23}: Continue factorization of 15: 15=3×515 = 3 \times 5
STEP_{24}: Combine the factors: 180=22×32×5180 = 2^2 \times 3^2 \times 5
STEP_{25}: Determine if 47 is prime. Since 47 has no divisors other than 1 and itself, it is prime.
STEP_{26}: Prime factorization of 88: 88=2×4488 = 2 \times 44
STEP_{27}: Continue factorization of 44: 44=2×2244 = 2 \times 22
STEP_{28}: Continue factorization of 22: 22=2×1122 = 2 \times 11
STEP_{29}: Combine the factors: 88=23×1188 = 2^3 \times 11
STEP_{30}:0 Prime factorization of 49: 49=7×749 = 7 \times 7
STEP_{31}:1 Determine if 83 is prime. Since 83 has no divisors other than 1 and itself, it is prime.
STEP_{32}:2 Prime factorization of 142: 142=2×71142 = 2 \times 71
Solution:
1. 42=2×3×742 = 2 \times 3 \times 7
2. 104=23×13104 = 2^3 \times 13
3. 75=3×5275 = 3 \times 5^2
4. 2323 is prime.
5. 27=3327 = 3^3
6. 72=23×3272 = 2^3 \times 3^2
7. 180=22×32×5180 = 2^2 \times 3^2 \times 5
8. 4747 is prime.
9. 88=23×1188 = 2^3 \times 11
10. 49=7249 = 7^2
11. 8383 is prime.
12. 142=2×71142 = 2 \times 71

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