An algebra problem by Paul Fournier

Algebra Level 2

An integer of two digits is N N times the sum of its digits. The number formed by interchanging the two digits is M M times the sum its digits. What is the value of M M ?

11-N 1+N N-1 9-N 10-N

Paul Fournier by Paul Fournier , 70, Canada

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1 solution

Eamon Gupta
Jun 24, 2015

Let the 2 digit integer be a b ‾ \overline{ab} , then we can create the following equations:

10 a + b = N ( a + b ) . . . . . . . . . ( 1 ) 10a+b=N(a+b).........(1)

10 b + a = M ( a + b ) . . . . . . . . ( 2 ) 10b+a=M(a+b)........(2)

Adding (1) and (2) we get 11 a + 11 b = ( M + N ) ( a + b ) 11a+11b=(M+N)(a+b) which canbe factorised:

11 ( a + b ) = ( M + N ) ( a + b ) 11(a+b)=(M+N)(a+b)

M + N = 11 M+N=11

Therefore M = 11 − N M=\boxed{11-N}

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