How can we show that the number of pairs ( a , b ) (where the pairs ( a

Simone Werner

Simone Werner

Answered question

2022-05-22

How can we show that the number of pairs ( a , b ) (where the pairs ( a , b ) and ( b , a ) are considered same) with lcm ( a , b ) = n is equal to
( 2 e 1 + 1 ) ( 2 e 2 + 1 ) . . . ( 2 e k + 1 ) + 1 2 ,
Where
n = p 1 e 1 p 2 e 2 p k e k , pis are prime for all 1 i k.

Answer & Explanation

bideanbarrenaf5

bideanbarrenaf5

Beginner2022-05-23Added 8 answers

First count the pairs with gcd ( a , b ) = n counting ( a , b ) and ( b , a ) as distinct. You have a = p i r i and b = p i s i and need max ( r i , s i ) = e i for all e. So, given a positive integer i, how many pairs ( r , s ) of nonnegative integers are there with max ( r , s ) = e?

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