QuestionWhich pair of numbers has an LCM of 60? 2 and 12 5 and 12 6 and 12 3 and 12
Studdy Solution
STEP 1
1. The Least Common Multiple (LCM) of a pair of numbers is the smallest positive integer that is divisible by both numbers.
2. To find the LCM of two numbers, we can use their prime factorizations.
3. We can also use the relationship between LCM and the Greatest Common Divisor (GCD): .
4. We need to determine the LCM for each given pair and check if it equals 60.
STEP 2
1. Find the prime factorization of each number.
2. Use prime factorizations to determine the LCM for each pair.
3. Compare the LCMs to see which pair has an LCM of 60.
STEP 3
Find the prime factorization of each number in the pairs.
\begin{align*}
2 &= 2 \\
12 &= 2^2 \cdot 3 \\
5 &= 5 \\
6 &= 2 \cdot 3 \\
3 &= 3 \\
\end{align*}
STEP 4
Determine the LCM of the pair (2, 12).
\begin{align*}
\text{Prime factors of 2: } & 2 \\
\text{Prime factors of 12: } & 2^2, 3 \\
\text{LCM: } & 2^2 \cdot 3 = 12
\end{align*}
STEP 5
Determine the LCM of the pair (5, 12).
\begin{align*}
\text{Prime factors of 5: } & 5 \\
\text{Prime factors of 12: } & 2^2, 3 \\
\text{LCM: } & 2^2 \cdot 3 \cdot 5 = 60
\end{align*}
STEP 6
Determine the LCM of the pair (6, 12).
\begin{align*}
\text{Prime factors of 6: } & 2, 3 \\
\text{Prime factors of 12: } & 2^2, 3 \\
\text{LCM: } & 2^2 \cdot 3 = 12
\end{align*}
STEP 7
Determine the LCM of the pair (3, 12).
\begin{align*}
\text{Prime factors of 3: } & 3 \\
\text{Prime factors of 12: } & 2^2, 3 \\
\text{LCM: } & 2^2 \cdot 3 = 12
\end{align*}
STEP 8
Compare the LCMs to determine which pair has an LCM of 60.
\begin{align*}
\text{LCM of (2, 12): } & 12 \\
\text{LCM of (5, 12): } & 60 \\
\text{LCM of (6, 12): } & 12 \\
\text{LCM of (3, 12): } & 12 \\
\end{align*}
Solution: The pair of numbers that has an LCM of 60 is .
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