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Math Help - divisibility

  1. #1
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    divisibility

    Investigate the questions in parts a and b by considering a number of examples

    a. If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?

    b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?

    c. based on your examples, what do you think the answers to the questions in a and b should be?

    d. Based on your answer to part c, determine whether 3,321,297,402,348,516 is divisible by 12 without using a calculator or doing long division. Explain your reasoning.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by ihavvaquestion View Post
    Investigate the questions in parts a and b by considering a number of examples

    a. If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?

    b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?

    c. based on your examples, what do you think the answers to the questions in a and b should be?

    d. Based on your answer to part c, determine whether 3,321,297,402,348,516 is divisible by 12 without using a calculator or doing long division. Explain your reasoning.
    a. 6 is divisible by 6 and 2.
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  3. #3
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    im not sure i follow you
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  4. #4
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    Hello, ihavvaquestion!

    (a) If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?
    No . . .

    As chiph588@ pointed out, 6 is divisible by 6 and by 2,
    . . but it is not divisible by 12.




    b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?
    Yes . . .


    \begin{array}{ccc}\text{Multiples of 3:} & 3,6,9,12,15,18,21,24,27,30,33,36,\hdots \\<br />
\text{Multiples of 4:} & 4,8,12,16,20,24,28,32,36,40,44,48,\hdots\end{array  }

    \begin{array}{ccc}\text{Multiples of 3 and 4:} & 12,24,36,48,60,72,84,\hdots<br />
\end{array}


    If a number is multiple of 3 and a multiple of 4, it is a multiple of 12.




    c. Based on your examples, what do you think
    the answers to the questions in (a) and (b) should be?
    Um . . . "No" and "Yes" ?



    d. Based on your answer to part (c), determine whether N = 3,\!321,\!297,\!402,\!348,\!516
    . . is divisible by 12 without using a calculator or doing long division.
    . . Explain your reasoning.

    A number is divisible by 3 if the sum of its digits is divisible by 3.

    Since 3+3+2+1+2+9+7 +4+0+2+3+4+8+5+1+6 \,=\,60,
    . . which is divisible by 3, then N is divisible by 3.


    A number is divisible by 4 if its last two-digit number is divisible by 4.

    Since N ends in 16, which is divisible by 4, then N is divisible by 4.


    Therefore, N is divisible by both 3 and 4 . . . N is divisible by 12.

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  5. #5
    Senior Member Tinyboss's Avatar
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    In general, you can say that "if x is divisible by a and b, then x is divisible by a*b" if and only if if a and b are coprime.
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  6. #6
    Super Member Bacterius's Avatar
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    Quote Originally Posted by Tinyboss
    In general, you can say that "if x is divisible by a and b, then x is divisible by a*b" if and only if if a and b are coprime.
    More accurately, if a | x and b | x then \textrm{lcm}{(a, b)} | x (and when a and b are coprime we have \textrm{lcm}{(a, b)} = a \times b).
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