Number elimination

Number Theory Level 3

The numbers from 1 to 500 are written on a piece of paper. In a single move you choose 2, 3, 4 or 5 numbers from the list, erase them and adjoin to the list the remainder of the sum of the chosen numbers when it is divided by 13. After a number of moves there are only two numbers left on the page. One of them is 102. What is the other?

256 10 12 20

Steve Iroghama by Steve Iroghama , 17, Nigeria

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2 solutions

  • x + 102 = 1 + 2 + 3 + . . . + 500 ( m o d 13 ) x+102=1+2+3+...+500(mod 13)
  • x + 102 = 500 ∗ 501 / 2 ( m o d 13 ) x+102=500*501/2(mod 13)
  • x + 102 = 3 ∗ 7 ( m o d 13 ) x+102=3*7(mod 13)
  • x + 11 = 21 ( m o d 13 ) x+11=21(mod 13)
  • x = 21 − 11 ( m o d 13 ) x=21-11(mod 13)
  • x = 10 ( m o d 13 ) x=10(mod 13)
1 Helpful 0 Interesting 0 Brilliant 0 Confused
Steve Iroghama
Feb 2, 2017

x+102=1+2+3+...+500(mod 13)

x+102=500.501/2(mod 13)

x+102=3.7(mod 13)

x+11=21(mod 13)

x=21-11(mod 13)

x=10(mod 13)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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