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Number Theory Level 3

Consider two positive integers a and b which are such that $a^{a}.b^{b}$ is divisible by 2000. What is the least possible value of the product $ab$ ?

(The question appeared in RMO 2000.)

1 solution

Harshit Joshi
Nov 15, 2014

Since 2000 divides $a^{a} \times b^{b}$ , it follows that 2 divides a or b and similarly 5 divides a or b.

In any case 10 divides ab. Thus the least possible value of ab for which 2000 divides $a^{a} \times b^{b}$ must be a multiple of 10.

Since 2000 divides $10^{10} \times 1^{1}$ , we can take a = 10,b = 1 to get the least value of ab equal to $\boxed{10}$ .

i did it in the same way

Harshita Moondra - 6 years, 6 months ago