Value of m and n

Question

If !!2^(3m+1)+3^(2n)+ 5^m+ 5^n = 2003!!, then !!(m,n)=!! ?

(a) (5,5)

(b) (3,3)

(c) (6,6)

(d) (10,10)

Solution

!!2^(3m+1)+3^(2n)+ 5^m+ 5^n = 2003!!

Here,

!!2^(3m+1) < 2003 < 2048= 2^11!!

!!=> 3m+1 < 11!!

!!=> m < 3.5!! or !! m <= 3 ......(1)!!

Clearly, !!3^(2n) < 2003 < 2187 = 3^7!!

!!=> 2n < 7!!

!!=> n < 3.5 !! or !!n <= 3!!

On putting !!(m,n) = (3,3)!!, the above equation satisfied.

Hence, (b) (3,3) is the answer.

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