Find the LCM and GCD of [tex]$2^3 \times 3^2 \times 5 \times 7$[/tex] and [tex]$2^2 \times 3^2 \times 5^2 \times 7$[/tex] and leave your answer in power form.

Find the GCD by taking the minimum exponent of each common prime factor: GC D = 2 min ( 3 , 2 ) × 3 min ( 2 , 2 ) × 5 min ( 1 , 2 ) × 7 min ( 1 , 1 ) .
Calculate the minimum exponents: GC D = 2 2 × 3 2 × 5 1 × 7 1 .
Find the LCM by taking the maximum exponent of each prime factor: L CM = 2 ma x ( 3 , 2 ) × 3 ma x ( 2 , 2 ) × 5 ma x ( 1 , 2 ) × 7 ma x ( 1 , 1 ) .
Calculate the maximum exponents: L CM = 2 3 × 3 2 × 5 2 × 7 1 .
The GCD is 2 2 × 3 2 × 5 × 7 ​ and the LCM is 2 3 × 3 2 × 5 2 × 7 ​ .

Explanation

Understanding the Problem We are given two numbers in their prime factorizations: 2 3 × 3 2 × 5 × 7 and 2 2 × 3 2 × 5 2 × 7 . We need to find their greatest common divisor (GCD) and least common multiple (LCM), expressing the results in power form.

Finding the GCD To find the GCD, we take the minimum exponent of each common prime factor. The prime factors are 2, 3, 5, and 7. Thus, the GCD is given by: GC D = 2 min ( 3 , 2 ) × 3 min ( 2 , 2 ) × 5 min ( 1 , 2 ) × 7 min ( 1 , 1 )

Calculating the GCD Calculating the exponents:



min ( 3 , 2 ) = 2
min ( 2 , 2 ) = 2
min ( 1 , 2 ) = 1
min ( 1 , 1 ) = 1 So, the GCD is: GC D = 2 2 × 3 2 × 5 1 × 7 1 = 2 2 × 3 2 × 5 × 7


Finding the LCM To find the LCM, we take the maximum exponent of each prime factor present in either number. The prime factors are 2, 3, 5, and 7. Thus, the LCM is given by: L CM = 2 ma x ( 3 , 2 ) × 3 ma x ( 2 , 2 ) × 5 ma x ( 1 , 2 ) × 7 ma x ( 1 , 1 )

Calculating the LCM Calculating the exponents:



ma x ( 3 , 2 ) = 3
ma x ( 2 , 2 ) = 2
ma x ( 1 , 2 ) = 2
ma x ( 1 , 1 ) = 1 So, the LCM is: L CM = 2 3 × 3 2 × 5 2 × 7 1 = 2 3 × 3 2 × 5 2 × 7


Final Answer Therefore, the GCD is 2 2 × 3 2 × 5 × 7 and the LCM is 2 3 × 3 2 × 5 2 × 7 .

Examples
Understanding LCM and GCD is crucial in many real-life scenarios. For instance, imagine you are tiling a floor with rectangular tiles. The GCD helps determine the largest square tile that can perfectly fit into the rectangular floor without any cuts. On the other hand, the LCM is useful when scheduling events that occur at different intervals. If you have two events, one happening every 12 days and another every 18 days, the LCM tells you when both events will occur on the same day, helping you coordinate schedules effectively. These concepts are fundamental in resource optimization and planning.

Answered by GinnyAnswer | 2025-07-05

The GCD of the numbers is 2 2 × 3 2 × 5 × 7 and the LCM is 2 3 × 3 2 × 5 2 × 7 .
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Answered by Anonymous | 2025-07-07