Find the L.C.M and G.C.D 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

The G.C.D. of the numbers is expressed as 2 2 × 3 2 × 5 × 7 while the L.C.M. is given as 2 3 × 3 2 × 5 2 × 7 . These results are computed by taking minimum and maximum powers of each common prime factor, respectively.
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Answered by Anonymous | 2025-07-05

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 . GC D = 2 2 × 3 2 × 5 × 7 , L CM = 2 3 × 3 2 × 5 2 × 7 ​

Explanation

Problem Analysis We are given two numbers, 2 3 × 3 2 × 5 × 7 and 2 2 × 3 2 × 5 2 × 7 , and 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 common 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 in Power Form Calculating the minimum exponents:



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


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 in Power Form Calculating the maximum exponents:



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


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 GCD and LCM is very useful in real life. For example, imagine you are tiling a floor. You want to use the largest square tile possible to cover a rectangular area without cutting any tiles. The side length of the largest square tile you can use is the GCD of the length and width of the rectangle. Similarly, if you have two events that occur regularly, like a bus arriving every 15 minutes and another every 20 minutes, the LCM tells you when they will both arrive at the same time. These concepts help optimize resources and planning in various scenarios.

Answered by GinnyAnswer | 2025-07-05