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: $$60=\frac{12\cdot b}{6}⟹60=2b⟹b=30$$

3. If GCD(12, b) = 4: $$60=\frac{12\cdot b}{4}⟹60=3b⟹b=20$$

4. If GCD(12, b) = 3: $$60=\frac{12\cdot b}{3}⟹60=4b⟹b=15$$

5. If GCD(12, b) = 2: $$60=\frac{12\cdot b}{2}⟹60=6b⟹b=10$$

6. If GCD(12, b) = 1: $$60=\frac{12\cdot b}{1}⟹60=12b⟹b=5$$

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 $\text{LCM}(12,b)=60$.