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Number Theory Level 2

Find the smallest positive integer which when divided by

  • 9 leaves remainder 8,
  • 8 leaves remainder 7,
  • 7 leaves remainder 6,
  • 6 leaves remainder 5,
  • 5 leaves remainder 4,
  • 4 leaves remainder 3,
  • 3 leaves remainder 2, and
  • 2 leaves remainder 1.


The answer is 2519.

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Rahul Singh by Rahul Singh , 21, India

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2 solutions

Mridul K Ojha
Jun 12, 2015

Here we have to take the LCM of all the numbers from 1 to 9.

i.e 2520.

Finally subtract the number by 1 to get the required solution.

THIS IS BECAUSE THE REQUIRED NUMBER IS JUST LESS THAN THE MULTIPLE OF NUMBERS ( 1 TO 9 ) BY 1

11 Helpful 2 Interesting 1 Brilliant 1 Confused
Paul Cockburn
Jun 16, 2018

The LCM of the numbers 2 to 9 is 2520 (= 9 x 8 x 7 x 5) and subtracting 1 will give a number which generates the required remainders. Hence 2519 is a possible answer. But how do we know there is not a smaller positive integer with the same property? Because if there was a smaller solution (call it n) then the integer n+1 would be a multiple of every number from 2 to 9, contradicting the fact that the LCM is 2520.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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