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Number Theory Level 3

What is the smallest positive integer that leaves a remainder of 1 when divided by 2, remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and so on up to a remainder of 9 when divided by 10?


The answer is 2519.

Harsh Shrivastava by Harsh Shrivastava , 20, India

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2 solutions

Ryan Tamburrino
Jul 28, 2014

Call the number n. Given the conditions, n+1 is a multiple of the numbers 1-10. The LCM of these numbers is 2520. Thus, the answer is 2519.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

it is good question.

Tatikonda Ravi Kishore - 6 years, 10 months ago

359 works as well.

William Isoroku - 6 years, 5 months ago
Harshit Joshi
Oct 26, 2014

As stated N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10

N + 1 must be a multiple of 2

N + 1 must be a multiple of 4 but if it is a multiple of 2 & 4 it is necessarily a multiple of 8

N + 1 must be a multiple of 5 but if it is a multiple of 2 & 5 it is necessarily a multiple of 10

N + 1 must be a multiple of 6 but if it is a multiple of 2 & 9 (18) it is necessarily a multiple of 6

N + 1 must be a multiple of 7

N + 1 must be a multiple of 9

Hence 2 x 4 x 5 x 7 x 9 = 2520 is a multiple of 2, 3, 4, 5, 6, 7, 8, 9 & 10

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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