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Math Help - Is there any easy way to find the remainders of very large integer divisions?

  1. #1
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    Is there any easy way to find the remainders of very large integer divisions?

    Hi all, I am Chaitanya, I am preparing for GRE, unfortunately I am poor at mathematics. But with some help I can understand quickly.

    I came across few questions, solved few on mu own. But I came across one question,

    What is the remainder of the division of 14414*14416*14418 by 14?

    Is there any easy way to solve this?

    Thank you all in advance and sorry if I ask a very silly question.
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  2. #2
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    It's called modular arithmetic. Look it up. It's quite useful, and will open you up to some interesting mathematics.
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  3. #3
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    Than you dude, do you mean I have to google "modular arithmetic"?
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  4. #4
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    Quote Originally Posted by krishna3264 View Post
    Than you dude, do you mean I have to google "modular arithmetic"?
    Yes, and you should start by looking at the first hit.

    modular arithmetic - Google Search

    CB
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  5. #5
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    I am reading from this page.

    I dint understood this para.
    Modular arithmetic lets us state these results quite precisely, and it also provides a convenient language for similar but slightly more complex statements. In the above example, our modulus is the number 2. The modulus can be thought of as the number of classes that we have broken the integers up into. It is also the difference between any two "consecutive" numbers in a given class.
    Can anyone explain? Thanks in advance.
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  6. #6
    Junior Member sorv1986's Avatar
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    we have to find the solution of the eqn
    x≡14414*14416*14418 (mod 14)

    When 14414 is divided by 14 , it leaves remainder as 8,
    14416 is divided by 14, it leaves remainder as 10,
    14418 is divided by 14, it leaves remainder as 12.
    so we get
    14414*14416*14418≡ 8*10*12≡80*12≡10*12≡8(mod 14)

    Hence,
    the remainder is 8 when 14414*14416*14418 is divided by 14.



    i think this will help.but please check the calculations
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  7. #7
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by krishna3264 View Post
    Hi all, I am Chaitanya, I am preparing for GRE, unfortunately I am poor at mathematics. But with some help I can understand quickly.

    I came across few questions, solved few on mu own. But I came across one question,

    What is the remainder of the division of 14414*14416*14418 by 14?

    Is there any easy way to solve this?

    Thank you all in advance and sorry if I ask a very silly question.
    even if you don't know modular arithmetic you can do this:
    14414=14k+8, \, 14416=14l+10, \, 14418=14m+12 \Rightarrow 14484 \cdot 14416 \cdot 14418=(14k+8)(14l+10)(14m+12).
    Expand this thing to get   14484 \cdot 14416 \cdot 14418=14p+8 \cdot 10 \cdot 12=14q+8, where p,q are integers.
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  8. #8
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    Thanks dude. I thought the answer would be 3. In the page I was reading(this) the author was describing about congruency for addition and multiplication. I thought for division

    0/0≡1mod2
    0/1≡0mod2
    1/0≡0mod2
    1/1≡1mod2

    Since even*even*even/even is even/even and 0/0≡1mod2, I thought answer is 3. Since the options given were
    8
    3
    12
    10
    6

    Since 3 is the only odd number I thought 3 would be the right answer. How ever there is little confusion in your post. How did you exactly said that 8 is the right answer. It can be either 10 or 12. Could you please explain.

    Thank you in advance
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  9. #9
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    Quote Originally Posted by sorv1986 View Post
    14, it leaves remainder as 12.
    so we get
    14414*14416*14418≡ 8*10*12≡80*12≡10*12≡8(mod 14)
    Hi Sorv1986, how did you say that 8 is the right answer. Could you please explain
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  10. #10
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by krishna3264 View Post
    Thanks dude. I thought the answer would be 3. In the page I was reading(this) the author was describing about congruency for addition and multiplication. I thought for division

    0/0≡1mod2 i don't what do you mean by this. 0/0????
    0/1≡0mod2
    1/0≡0mod2
    1/1≡1mod2

    Since even*even*even/even is even/even and 0/0≡1mod2, I thought answer is 3. Since the options given were
    8
    3
    12
    10
    6

    Since 3 is the only odd number I thought 3 would be the right answer. How ever there is little confusion in your post. How did you exactly said that 8 is the right answer. It can be either 10 or 12. Could you please explain.

    Thank you in advance
    ...
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  11. #11
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    @abhishek, I just thought divisions will be represented like that. My guess was wrong.

    All I want to know is, how you people are telling 8 is the right answer?

    Thank you in advance.
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  12. #12
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by krishna3264 View Post
    @abhishek, I just thought divisions will be represented like that. My guess was wrong.

    All I want to know is, how you people are telling 8 is the right answer?

    Thank you in advance.
    i have solved it in post #7. what step is not clear to you?
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  13. #13
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    Is modular arithimetic just a way of finding remainders?
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  14. #14
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    It's actually quite a bit more than that, but that is something that it can be used for.
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  15. #15
    Junior Member sorv1986's Avatar
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    Quote Originally Posted by krishna3264 View Post
    Hi Sorv1986, how did you say that 8 is the right answer. Could you please explain
    a≡b(mod n) if and only if n divides (b-a).

    so when 80 is divided by 14, the remainder will be 10 .(as 80=14.5+10)

    similarly the rests.

    would like to know which part of the solution is foggy?
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