Find the greatest 4-digit number, which is a perfect square. Find its square root.
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Solution
The greatest 4-digit number is 9999.
First, find the least number which when substracted from 9999 gives a perfect square. ∣99¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯9∣99–––99–––∣−81¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯189∣1899∣−1701¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣198 The remainder is 198. The required number, which is a perfect square is 9999−198=9801 and clearly, √9801=99.