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Given the prime factorization of an integer how can you determine if our integer is a perfect square?
4, 16, 64 and many others
49 x 14 x 18 is not a perfect square. This is because a prime factorization of 49 x 14 x 18 contains 3 sevens. The prime factorization of any perfect square must have an even amount of each prime number.
16.
16
16 2*16 = 32
144 is a perfect square, whose square factor is 12: Therefore we simply repeat the prime factorization of 12 (3,2,2) twice: 2,2,2,2,3,3.
The square root of 512 is neither an integer, nor even a rational number, so it has no prime factorization.
One way is to get the prime factorization of the number. If every prime occurs an even number of times, it is a square, otherwise, not. Another is to estimate the square root of the number, and square it. If you get more than the number, try a lower estimate; if less, a higher one. Using interval bisection you very quickly zero in on the square root, if it is a whole number. If so, the number is a perfect square. Otherwise, you find 2 consecutive whole numbers between which is the square root, in which case it is not a perfect square.
16
No.
Nice question! The square root of (any number that isn't a perfect square) is irrational. No prime number is a perfect square. So the square root of any prime number is irrational.