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.
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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.
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If GCD(12, b) = 6: \[ 60 = \frac{12 \cdot b}{6} \implies 60 = 2b \implies b = 30 \]
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If GCD(12, b) = 4: \[ 60 = \frac{12 \cdot b}{4} \implies 60 = 3b \implies b = 20 \]
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If GCD(12, b) = 3: \[ 60 = \frac{12 \cdot b}{3} \implies 60 = 4b \implies b = 15 \]
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If GCD(12, b) = 2: \[ 60 = \frac{12 \cdot b}{2} \implies 60 = 6b \implies b = 10 \]
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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 \( \text{LCM}(12, b) = 60 \).