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# Modular Math

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Find the smallest n that will satisfy the following modular equations:

n mod 1657 =1162,  n mod 1162 =453. Any help would be appreciated. Thank you.

Jul 19, 2017

#1
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*editing

We can rewrite the modular equations as:

\(n-1162=1657k,\\ n-453=1162m.\)

Now we are able to remove from the equation:

\(1657k+1162=1162m+453,\)

we can split the first term to tidy things up:

\(1162k+495k+1162-1162m=453\Rightarrow\\ 1162(k+1-m)+495k=453\Rightarrow\\ 1162(k+1-m)+495k-495=453-495\Rightarrow\\ 1162(k+1-m)+495(k-1)=-42.\)

The first term is even and the outcome is even, hence the second term must be even as well. We conclude that 495(k-1) is a multiple of 10. In order to get a negative number (k+1-m) must be negative as well. Since the second term doesn't contribute to the last digit of the sum we know the last digit of the first term must also be 2. As a result (k+1-m) must have a last digit equal to 1 or 6.

Jul 19, 2017
edited by Honga  Jul 19, 2017
edited by Honga  Jul 19, 2017
#2
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1657k+1162 =1162m + 453. By simple iteration:

k = 15 and  m =22. Therefore, the smallest n =1657*15+1162 =26,017

Since the LCM of 1657 and 1162 =1,925,434, therefore:

n = 1,925,434D + 26,017, where D =0, 1, 2, 3......etc.

Jul 19, 2017