Aptitude - Numbers

80 = 2 x 5 x 8

Since 653xy is divisible by 2 and 5 both, so y = 0.

Now, 653x is divisible by 8, so 13x should be divisible by 8.

This happens when x = 6.

x + y = (6 + 0) = 6.

Let the two consecutive odd integers be (2n + 1) and (2n + 3). Then,

(2n + 3)² - (2n + 1)² = (2n + 3 + 2n + 1) (2n + 3 - 2n - 1)

     = (4n + 4) x 2

     = 8(n + 1), which is divisible by 8.

Unit digit in (4137)⁷⁵⁴ = Unit digit in {[(4137)⁴]¹⁸⁸ x (4137)²}

=Unit digit in { 292915317923361 x 17114769 }

= (1 x 9) = 9

 4 | x            z = 6 x 1 + 4  = 10
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 5 | y -2         y = 5 x z + 3  = 5 x 10 + 3  = 53
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 6 | z - 3        x = 4 x y + 2  = 4 x 53 + 2  = 214
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   | 1 - 4
   
Hence, required number = 214.