Question

3.1

1. use powers to write each number as a product of its prime factors.

a) 1260
b) 4224
c) 6120
d) 1045
e) 3024
f) 3675

Explanation:

Step1: Prime factorize 1260

Divide by primes sequentially:
$1260 \div 2 = 630$
$630 \div 2 = 315$
$315 \div 3 = 105$
$105 \div 3 = 35$
$35 \div 5 = 7$
$7 \div 7 = 1$
So $1260 = 2^2 \times 3^2 \times 5 \times 7$

Step2: Prime factorize 4224

Divide by primes sequentially:
$4224 \div 2 = 2112$
$2112 \div 2 = 1056$
$1056 \div 2 = 528$
$528 \div 2 = 264$
$264 \div 2 = 132$
$132 \div 2 = 66$
$66 \div 2 = 33$
$33 \div 3 = 11$
$11 \div 11 = 1$
So $4224 = 2^7 \times 3 \times 11$

Step3: Prime factorize 6120

Divide by primes sequentially:
$6120 \div 2 = 3060$
$3060 \div 2 = 1530$
$1530 \div 2 = 765$
$765 \div 3 = 255$
$255 \div 3 = 85$
$85 \div 5 = 17$
$17 \div 17 = 1$
So $6120 = 2^3 \times 3^2 \times 5 \times 17$

Step4: Prime factorize 1045

Divide by primes sequentially:
$1045 \div 5 = 209$
$209 \div 11 = 19$
$19 \div 19 = 1$
So $1045 = 5 \times 11 \times 19$

Step5: Prime factorize 3024

Divide by primes sequentially:
$3024 \div 2 = 1512$
$1512 \div 2 = 756$
$756 \div 2 = 378$
$378 \div 2 = 189$
$189 \div 3 = 63$
$63 \div 3 = 21$
$21 \div 3 = 7$
$7 \div 7 = 1$
So $3024 = 2^4 \times 3^3 \times 7$

Step6: Prime factorize 3675

Divide by primes sequentially:
$3675 \div 3 = 1225$
$1225 \div 5 = 245$
$245 \div 5 = 49$
$49 \div 7 = 7$
$7 \div 7 = 1$
So $3675 = 3 \times 5^2 \times 7^2$

Answer:

a) $2^2 \times 3^2 \times 5 \times 7$
b) $2^7 \times 3 \times 11$
c) $2^3 \times 3^2 \times 5 \times 17$
d) $5 \times 11 \times 19$
e) $2^4 \times 3^3 \times 7$
f) $3 \times 5^2 \times 7^2$