4, 16, 64 and many others
16
Given the prime factorization of an integer how can you determine if our integer is a perfect square?
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
To determine the least number that 5202 must be multiplied by to become a perfect square, we first need its prime factorization. The prime factorization of 5202 is (2 \times 3 \times 7 \times 37). For a number to be a perfect square, all primes in its factorization must have even exponents. Here, each prime has an exponent of 1, so we need to multiply by each prime factor once more: (2 \times 3 \times 7 \times 37 = 5202). Therefore, the least number to multiply 5202 by is 5202.
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.
16
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.
No.