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Puzzlette: How many multiples?

November 14, 2007 am30 7:06 am

Do you have a cute way of counting the multiples of a number on a given interval? Eg. how many multiples of 11 are there from 2000 to 2500? (that sort of question)

Came up in the comments (here) and it occurred to me that I do a lot of things without thinking about them.  I think I am practical, but not cute. I wonder what you do. Tell us.

from → Math, Math Education, mathematics, Puzzles
2 Comments leave one →
  1. mathmom permalink
    November 14, 2007 am30 9:07 am 9:07 am

    What I do is find a multiple of the number (i.e. 11 in this case) near the start and end of the range in question. 11 has a nice divisibility rule, and it’s easy to see that 2002 and 2508 are multiples of 11. (For anyone who doesn’t know the rule, google it.) ;-)

    Then my thinking goes that from 2003 through 2508 I have a number of complete “sets” modulo 11 — that is I am starting from a number one more than a multiple of 11, then 2 more, … 10 more and finally a multiple of 11. Each of these sequences has 11 numbers and one of them, the last, is a multiple of 11. There are 506 numbers from 2003 to 2508 inclusive, and 506/11 = 46. (Good, 506 is as expected also a multiple of 11, since I have a number of complete 11-number sequences, so I didn’t mess up my subtraction…) So there are 46 multiples of 11 in [2003, 2508]. The last multiple, 2508 is outside of the desired range, so we take it out. Likewise we add back in 2002 that we left off. So there are 46 multiples of 11 in the interval [2000,2500].

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  2. vlorbik permalink
    November 14, 2007 pm30 6:02 pm 6:02 pm

    break out the calculator.
    2000/11 = 181.8181818181….
    so 182*11 (=2002) is the least
    multiple of eleven in the given range.
    likewise 2500/11 = 227.27272727…
    so 227*11 (= 2497) is the greatest.
    our answer is now 227 – 182 + 1 =46.
    the “+1” of course accounts for the
    notorious “fencepost error”
    (wikipedia; jargon file).

    [explicitly, one has
    2002 = 2002 + 0*11
    2013 = 2002 + 1*11
    2024 = 2002 + 2*11
    .
    .
    .
    2497 = 2002 +45*11;
    the presence of “0” in the first line
    accounts for the need for “+1”
    in this context.]

    same as mathmom, in other words,
    but with calculator in hand and
    bypassing all the sanitiy checking.
    one has played this game before …

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