Question

When a square number ends in 6,its square root will have 6 in its unit's place.how?

Answer 1

Not always, but can be. If a perfect square ends in 6 then it's square root will end either in 4 or in 6. Let me prove the converse by Euclid's Division Lemma.

Consider a number ending in 4,

a = 10q + 4

Squaring both sides,

a² = (10q + 4)²

a² = 100q² + 80q + 16

a² = 100q² + 80q + 10 + 6

a² = 10(10q² + 8q + 1) + 6

a² = 10m + 6, where m = (10q² + 8q + 1).

Thus the square ends in 6.

Consider another number ending in 6.

a = 10q + 6

As earlier, squaring both sides,

a² = (10q + 6)²

a² = 100q² + 120q + 36

a² = 100q² + 120q + 30 + 6

a² = 10(10q² + 12q + 3) + 6

a² = 10n + 6, where n = (10q² + 12q + 3).

Thus the square of this number also ends in 6.

In case of other numbers ending in any other number except 4 and 6, we can't see their squares ending in 6.

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