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Doubt & Answer Explanation

Doubt

Respected faculty
I am not able to get the explanation given for questionregarding power of 82 in 83!-52!.please elaborate this

Answer Explanation

For getting the power of 82 in 83! -52!
we have to find the power of 41 and 2.

Since there are less power of 41 in this, so we will calculate it first.
Also power of 41 will be equal to power of 82 since there are large power of 2 in 83! -52!

Now,
83! - 52! = 52! (83x82x81x80.......x54x53 - 1)

now

52! has one power of 41.
and A= 83x82x81...54X53 is divisible by 41( as 82 is multiple of 41)
therefore
A -1 is not divisible by 41 (since A is divisible by 41)

Therefore,
83! - 52! has only single power of 41.

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