Divisible by 5

Number Theory Level 3

If p = 6 m + 9 n p=6^m+9^n where m , n ∈ ( 1 , 2 , 3 , … , 50 ) m,n \in (1,2,3,\ldots, 50) . Find the number of pairs of ( m , n ) (m,n) for which p p is divisible by 5.


The answer is 1250.

Md Zuhair by Md Zuhair , 20, India

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1 solution

Kushal Bose
Dec 20, 2016

p = 6 m + 9 n = ( 5 + 1 ) m + ( 10 − 1 ) n = 5 k + ( 1 ) m + ( − 1 ) n p=6^m+9^n=(5+1)^m + (10-1)^n=5 k + (1)^m +(-1)^n

To be divisible by 5 5 here m m can be any integer from the set and n n should be odd.

So pairs of ( m , n ) (m,n) are 50 × 25 = 1250 50 \times 25=1250

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