A positive integer is a perfect square only if every exponent in its prime factorization is even (choose any whole number at all, prime factorize it, square it, and you should see why). Neither statement comes close to giving us this information:
1) k is divisible by 2^2. k might be (2^2)*3, or 2^3, or a lot of other numbers which are not perfect squares. Not sufficient.
2) k is divisible by four different primes. k might be 2*3*5*7 (or, again, a lot of other numbers which are not perfect squares). Not sufficient.
Even together, k might be (2^2)*3*5*7 (or a lot of other non-squares), so not sufficient.
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Ian Stewart
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Ian Stewart
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Divide into primes. For example:vinviper1 wrote:what do you mean by prime factorize it?
45 = (3^2)*5
72 = (2^3)*(3^2)
120 = (2^3)*3*5
Prime factorizations are very useful in almost every question about divisibility.