A number theory problem by Ayush G Rai

Number Theory Level 3

d d is an integer greater than 1. When the numbers 1059,1417 and 2312 are divided by d d , the remainders in each case is the same r r . Find the value of d − r d-r .


The answer is 15.

Ayush G Rai by Ayush G Rai , 20, India

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2 solutions

Razzi Masroor
Feb 13, 2017

1059 = r mod d, 1417= r mod d and 2312= r mod d.So 1417-1059= 0 mod d, 2312-1417=0 mod d, and 2312-1059=0 mod d.This means that the numbers 358, 1253, and 895 are divisible by d. 358 is equal to 179 times 2, 895 is equal to 5 times 179 and 1253= 7 times 179 so d=179.Then you can just find the remainder of 1059 or 1417 or 2312 when divided by 179.The remainder is 164 so d-r is 15.

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Roger Erisman
May 10, 2016

For i = 1 To 1059 

a = math.remainder(1059,i)

b = math.remainder(1417,i)

c = math.remainder(2312,i)

If a = b Then

  If b = c Then

    TextWindow.Write(i)

    TextWindow.Write(" ")

    TextWindow.WriteLine(a)

  EndIf

EndIf

EndFor

Results: 1 0

179 164

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I did not understand your solution.

Ayush G Rai - 5 years ago

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