Question

1. what are the common factors of 12 and 24? of 12 and 36?
2. what is the greatest common factor of 9 and 12? of 24 and 36?
3. list the first twelve nonzero multiples of 3, 6, and 9.
4. using the answer to problem 33, circle the common multiples of 3 and 9 in green. circle the common multiples of 3 and 6 in red.

Explanation:

Problem 31

Step1: Find factors of 12

The factors of 12 are all numbers that divide 12 without leaving a remainder. So, \(12\div1 = 12\), \(12\div2 = 6\), \(12\div3 = 4\), \(12\div4 = 3\), \(12\div6 = 2\), \(12\div12 = 1\). Thus, factors of 12: \(1, 2, 3, 4, 6, 12\).

Step2: Find factors of 24

For 24, \(24\div1 = 24\), \(24\div2 = 12\), \(24\div3 = 8\), \(24\div4 = 6\), \(24\div6 = 4\), \(24\div8 = 3\), \(24\div12 = 2\), \(24\div24 = 1\). Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\).

Step3: Find common factors of 12 and 24

Common factors are the numbers present in both factor lists. So, common factors of 12 and 24: \(1, 2, 3, 4, 6, 12\).

Step4: Find factors of 36

For 36, \(36\div1 = 36\), \(36\div2 = 18\), \(36\div3 = 12\), \(36\div4 = 9\), \(36\div6 = 6\), \(36\div9 = 4\), \(36\div12 = 3\), \(36\div18 = 2\), \(36\div36 = 1\). Factors of 36: \(1, 2, 3, 4, 6, 9, 12, 18, 36\).

Step5: Find common factors of 12 and 36

Common factors of 12 and 36 are numbers in both factor lists of 12 and 36. So, common factors of 12 and 36: \(1, 2, 3, 4, 6, 12\).

Step1: Find GCF of 9 and 12

Factors of 9: \(1, 3, 9\) (since \(9\div1 = 9\), \(9\div3 = 3\), \(9\div9 = 1\)). Factors of 12: \(1, 2, 3, 4, 6, 12\) (from Problem 31). Common factors of 9 and 12: \(1, 3\). The greatest among them is 3.

Step2: Find GCF of 24 and 36

Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\) (from Problem 31). Factors of 36: \(1, 2, 3, 4, 6, 9, 12, 18, 36\) (from Problem 31). Common factors of 24 and 36: \(1, 2, 3, 4, 6, 12\). The greatest among them is 12.

Step1: Multiples of 3

A multiple of a number is obtained by multiplying it with positive integers. For 3: \(3\times1 = 3\), \(3\times2 = 6\), \(3\times3 = 9\), \(3\times4 = 12\), \(3\times5 = 15\), \(3\times6 = 18\), \(3\times7 = 21\), \(3\times8 = 24\), \(3\times9 = 27\), \(3\times10 = 30\), \(3\times11 = 33\), \(3\times12 = 36\). So first twelve non - zero multiples of 3: \(3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36\).

Step2: Multiples of 6

For 6: \(6\times1 = 6\), \(6\times2 = 12\), \(6\times3 = 18\), \(6\times4 = 24\), \(6\times5 = 30\), \(6\times6 = 36\), \(6\times7 = 42\), \(6\times8 = 48\), \(6\times9 = 54\), \(6\times10 = 60\), \(6\times11 = 66\), \(6\times12 = 72\). First twelve non - zero multiples of 6: \(6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72\).

Step3: Multiples of 9

For 9: \(9\times1 = 9\), \(9\times2 = 18\), \(9\times3 = 27\), \(9\times4 = 36\), \(9\times5 = 45\), \(9\times6 = 54\), \(9\times7 = 63\), \(9\times8 = 72\), \(9\times9 = 81\), \(9\times10 = 90\), \(9\times11 = 99\), \(9\times12 = 108\). First twelve non - zero multiples of 9: \(9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108\).

Answer:

Common factors of 12 and 24: \(1, 2, 3, 4, 6, 12\); Common factors of 12 and 36: \(1, 2, 3, 4, 6, 12\)

Problem 32