Question

LCM of two integers is 60 and one of the integers is 12. Write down 2 possible values of the other integer.

Answer 1

To find the possible values of the other integer \( b \) given that the least common multiple (LCM) of \( 12 \) and \( b \) is \( 60 \), we can use the relationship:

\[
\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}
\]

Where:
- \( a = 12 \)
- \( \text{LCM} = 60 \)

We can rearrange the formula to find \( b \):

\[
b = \frac{\text{LCM}(a, b) \cdot \text{GCD}(a, b)}{a}
\]

Now, we need to find possible values for \( b \) such that this relationship holds.

First, we can find the GCD of \( 12 \) and \( b \):
- The factors of \( 12 \) are \( 1, 2, 3, 4, 6, 12 \).
- The GCD must be one of these factors.

We'll calculate possible values for \( b \) using the values of the GCD.

1. **If GCD(12, b) = 12:**
\[
LCM(12, b) = \frac{12 \cdot b}{12} = b
\]
Setting \( b = 60 \) does not work since that's not equal to 12.

2. **If GCD(12, b) = 6:**
\[
60 = \frac{12 \cdot b}{6} \implies 60 = 2b \implies b = 30
\]

3. **If GCD(12, b) = 4:**
\[
60 = \frac{12 \cdot b}{4} \implies 60 = 3b \implies b = 20
\]

4. **If GCD(12, b) = 3:**
\[
60 = \frac{12 \cdot b}{3} \implies 60 = 4b \implies b = 15
\]

5. **If GCD(12, b) = 2:**
\[
60 = \frac{12 \cdot b}{2} \implies 60 = 6b \implies b = 10
\]

6. **If GCD(12, b) = 1:**
\[
60 = \frac{12 \cdot b}{1} \implies 60 = 12b \implies b = 5
\]

Thus, the possible values for \( b \) are \( 30, 20, 15, 10, \) and \( 5 \).

Two possible values of the other integer \( b \) are:

- \( 30 \)
- \( 20 \)

These will satisfy that \( \text{LCM}(12, b) = 60 \).