n ∈ N, b, a ∈ N ; a≠0. In base 10; n=aabb^(−) 1. show that n is not prime. 2. Give conditions on b such that n is perfect square. 3. Determinate n such that n is a perfect square.


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Question Number 142388 by mathocean1 last updated on 30/May/21

$n \in N, b, a \in N; a \neq 0 .$ $In base 10; n = \overset{―}{aabb}$ $1 . show that n is not prime .$ $2 . Give conditions on b such that$ $n is perfect square .$ $3 . Determinate n such that n is a$ $perfect square .$
	Answered by mr W last updated on 31/May/21

	$1 .$ $n = 1000 a + 100 a + 10 b + b$ $= 1100 a + 11 b$ $= 11 (100 a + b) \equiv 0 (m o d 11)$ $2 .$ $100 a + b = 11 m^{2} ⩾ 101 \Rightarrow m ⩾ 4$ $100 a + b = 11 m^{2} ⩽ 909 \Rightarrow m ⩽ 9$ $b = 11 m^{2} - 100 a$ $3 .$ $n = 11^{2} m^{2} = {(11 m)}^{2} w i t h 4 ⩽ m ⩽ 9$
	Commented by mathocean1 last updated on 31/May/21

	$S o r r y . . . p l e a s e s i r; a c c o r d i n g t o t h e$ $s e c o n d q u e s t i o n w h y h a v e y o u$ $100 a + b ⩽ 909 a n d ⩾ 101 ?$
	Commented by mr W last updated on 31/May/21

	$1 ⩽ a, b ⩽ 9$ $100 a + b ⩽ 100 \times 9 + 9 = 909$ $100 a + b ⩾ 100 \times 1 + 1 = 101$
	Commented by mathocean1 last updated on 31/May/21

	$t h a n k s$
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