Minimum possible value

If x x and y y are positive integers such that x x when divided by y y , leaves a remainder 29, what is the minimum possible value of x y xy ?


The answer is 870.

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1 solution

Akshat Jain
Jan 5, 2014

Since x x on division by y y leaves remainder 29 29 , we can write this situation mathematically as x = k y + 29 x=ky+29 where k k is a non-negative number. We have to find the minimum value of x y xy , which we can write as- x y = ( k y + 29 ) y xy=(ky+29)y = k y 2 + 29 y =ky^2+29y Now, to minimize the above reached expression, we must minimize k k and y y . Since k k is a non-negative number, it's minimum value is 0 0 . Also, we know that if a number (say, a a ) leaves a remainder r r on division by b b , that is, a = b q + r a=bq+r , then b > r b>r . Therefore, in this case, y > 29 y>29 . So the minimum possible integer value of y y is 30 30 . Now that we have all the values, we can put them in the expression to get the minimum value as- 0 × 3 0 2 + 29 30 = 29 × 30 = 870 0 \times30^2 + 29\cdot30 = 29 \times30 = \fbox{870} And we are done.

really?

Aashi Jain - 7 years, 4 months ago

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