Question

find the greatest number of 6 digits . which is a perfect square.

Answer 1

It's 1,000,000 which is 1000squared. Now try smaller numbers squared. 999 squared is 998001. There's no integer between 999 and 1000, so998001 is the largest perfect square with 6 digits.

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Answer 2

Let x² be the largest six-digit perfect square.

The largest six-digit number is 999,999, so
x² ≤ 999,999.

You know that 1,000² = 1,000,000, which is only 1 more than 999,999, so x² must be the next smaller square, which is (1,000 - 1)².
(1,000 - 1)² = 999² = 998,001

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