N-lt-aabb-gt-N-amp-N-is-perfect-square-find-N-

Question Number 192786 by MM42 last updated on 27/May/23

N =< a a b b >\in N & N i s p e r f e c t s q u a r e

f i n d N ?

Answered by AST last updated on 27/May/23

1100 a + 11 b = x^{2} \Rightarrow 11 (100 a + b) = x^{2}

11 ∣ 100 a + b \Rightarrow 100 a + b = 11 p^{2} \Rightarrow 11 ∣ a + b

(a, b) \neq (6, 5), b \neq 2, 3, 6, 7, 8 (a p e r f e c t s q u a r e c a n n o t

e n d i n 2, 3, 7 o r 8, p o w e r o f 2 i n \overset{____}{a a b b} w h e n b = 6 i s 1)

\Rightarrow P o s s i b l e v a l u e s o f (a, b) = (2, 9), (7, 4)

O f t h e s e, o n l y (7, 4) g i v e s a p e r f e c t s q u a r e

\Rightarrow N = 7744

Commented by BaliramKumar last updated on 27/May/23

perfect square number last two same digit only

00 or 44

∵ 11 ∣ a + b

∴ bb \neq 00

(if bb = 00 then 11 = a > 9, {it}^{'} s not possible)

Commented by MM42 last updated on 28/May/23

b r a v o

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