What's the number?

Number Theory Level pending

There exists a 2-digit positive integer x y ‾ \overline{xy} such that 2 ( 10 x + y ) = 10 ( y + 2 ) + x 2(10x + y) = 10(y+2) + x .

Find the value of this 2-digit integer.


The answer is 47.

Arnab Kumar Debnath by Arnab Kumar Debnath , 21, Bangladesh

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1 solution

As it is said that 2 ∗ ( 10 ∗ x + y ) = 10 ∗ ( y + 2 ) + x 2*(10*x+y)=10*(y+2)+x ,from here we can say that 19 ∗ x − 8 ∗ y + 20 = 0 19*x-8*y+20=0 . Now using the moduler math we can say that dividing 19 x by 8 would give us a reminder 4. So it means that dividing x by 8 would also give us a reminder 4.That means x = 8 ∗ t + 4 ) x=8*t+4) . As 0<x<10 ,(-1/2)<t<(3/4).As a result t will have only one integer and that is 0. So x=8 (0)+4=4 and y=7.The number stands 4*10+7=47

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