Question 129894This question is from textbook ELEMENTARY NUMBER THEORY
: Find an integer having the remainders 1,2,5,5, when divided by 2,3,6,12, respectively
This question is from textbook ELEMENTARY NUMBER THEORY
Found 2 solutions by Dalmo, Earlsdon:
Answer by Dalmo(2)   (Show Source):
You can put this solution on YOUR website! Hi there!
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For this problem, we don't need to use a whole lot of math. We just need to pay attention, think about it, and write down our steps.
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We know that it is an odd number, since its remainder is 1 when divided by 2.
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We also know that is is 2 more than an odd multiple of 3.
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Let's get some options now:
Multiples of 3: 0,3,6,9,12,15,18,21,24...
Odd multiples of 3: 3,9,15,21...
Options: 7,11,17,23...
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If we divide the number by 6, the remainder is 5.
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Options: 17,23
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If we divide the number by 12, the remainder is 5
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Answer: 17
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I hope it helps.
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Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website! Find an integer having remainders of 1, 2, 5, 5, when divided by 2, 3, 6, 12 respectively.
Well, intuitively, you can see that the required integer has to be equal to or larger than 12+5, right?
In fact, you can go farther and say that it can be equal to n*12+5, where n is any non-zero positive integer (1, 2, 3, ...).
Let's try some:
n = 1, so n*12+5 = 17
17/2 = 8 & R=1
17/3 = 5 & R=2
17/6 = 2 & R=5
17/12 = 1 & R=5
So 17 is such an integer and it happens to be the smallest such integer.
Here are some others that you can try:
2*12+5 = 29
3*12+5 = 41
4*12+5 = 53
5*12+5 = 65,...
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