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# Number Theory

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A very large number x is equal to $2^2 \cdot 3^3 \cdot 4^4 \cdot 5^5 \cdot 6^6$. What is the smallest positive integer that, when multiplied with x, produces a product that is a perfect square?

Oct 25, 2021

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x ==2^2×3^3×4^4×5^5×6^6

Look at the exponents of x. ANY exponent that is EVEN is already a perfect square. All we need then is to make the exponent of 3 and 5 EVEN by multiplying them by: 3 x 5 ==15, which is the smallest number that will make x a perfect square: 2^2  x  3^4  x 4^4  x  5^6  x  6^6=60,466,176,000,000.

Square root of [60,466,176,000,000]==7,776,000 - whixh is a perfect square.

Oct 25, 2021