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10 = 2*5 (prime decomposition)
So for a power of 10 to be a square means that the powers its factors must be even. A power of 10 is a square if it contains an even number of factors of 2 and 5. Since 503 is an odd number, 10^503 has an odd number of powers of 2 and 5, so it can't be the square of an integer.

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Q: Why cant 10 to the power of 50 3 be a square number?
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