Probability 1/8100 to guess it successfully
I am given 2 four-digit positive integers and such that:
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If is divided by the sum of its digits, it gives a quotient of 327 and remainder of 14.
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If is divided by the sum of its digits, it gives a quotient of 227 and remainder of 16.
Find the value of the integer .
The answer is 6554.
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1 solution
When you subtract reverse of a number from itself you get a multiple of 9 . Let S = x + y + z + w , it follows that A = 3 2 7 S + 1 4 and B = 2 2 7 S + 1 6 and A − B = 1 0 0 S − 2 . Take modulo 9 : 0 ≡ 9 A − B ≡ 9 1 0 0 S − 2 ≡ 9 S − 2 S − 2 ≡ 9 0 And since 0 ≤ S ≤ 3 6 we must have s ∈ { 2 , 1 1 , 2 0 , 2 9 } . It's easy to see that just S = 2 0 works and A = 6 5 5 4 .