The largest square factor of 557 is 1 because 557 is a prime number
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The long but fool-proof way is to find the complete prime factorisation of the number. Then group all the factors into pairs, rejecting any that cannot be paired. The product of the numbers that are left is the largest square factor.
A factor that is a square number. 16 is a square factor of 32.
The largest factor of any number is the number itself.