October 22, 2015

Peculiar Number Puzzle

I just found a number with an interesting property:
When I divide it by 2, the remainder is 1.
When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
When I divide it by 9, the remainder is 8.
When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.
Find the smallest number with such property.

5 comments:

  1. MovinOctober 22, 2015 at 4:01 PM

    Remainder on dividing by any K (K from 2 to 10) is K -1. So the number can be lcm(1,2,3,...10)-1 = 2519

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  2. UnknownOctober 22, 2015 at 4:37 PM

    119

    ReplyDelete
  3. Josh D.October 23, 2015 at 3:40 AM

    I believe the solution is 2,519.

    From the first statement, having a remainder of 1 means it is one less than an number divisible by 2.
    From the second statement, having a remainder of 2 means it is one less than a number divisible by 3.
    From the third statement, having a remainder of 3 means it is one less than an number divisible by 4.
    ..etc... for the rest of the statements.

    This means that it is one less than a number that is divisible by 2,3,4,5,6,7,8,9,10. That means we need the Least-Common-Multiple (LCM) of those nine numbers.

    Getting the prime factorization of each of those numbers.
    2 = 2^1
    3 = 3^1
    4 = 2^2
    5 = 5^1
    6 = 2^1 * 3^1
    7 = 7^1
    8 = 2^3
    9 = 3^2
    10 = 2^1 * 5^1

    The LCM is the product of the highest exponents for each of those bases.
    So the LCM in this case = 2^3 * 3^2 * 5^1 * 7^1 = 8*9*5*7 = 2,520.

    Our answer is one less than that number, so our answer is 2,519.

    You can do the division with each of those to check that the remainders are correct.

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  4. Suvinthra MOctober 23, 2015 at 1:03 PM

    I go with Josh D's answer and explanation : - 2519.

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  5. Jerry CritterOctober 23, 2015 at 9:47 PM

    Yes, 2519

    ReplyDelete