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2 Modulo Arithmetic Problems

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Hi, can someone help me with these problems? It'd mean so much to me:

1. How many positive integers N from 1 to 5000 satisfy the congruence $$N \equiv 11 \pmod{13}$$?

2. How many positive integers N from 1 to 5000 satisfy both congruences, $$N\equiv 5\pmod{12}$$ and $$N\equiv 11\pmod{13}$$?

Thank you so much!

Nov 14, 2020

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1 - N =13D  +  11, where D =0, 1, 2, 3........etc.

The smallest N = 11

5,000 - 11 =4,989

4,989 / 13 =383 - these are multiples of 13 beween 11 and 5000.

So you have a total of 383 + 1 =384 integers between 1 and 5000. They begin as follows:

[13 * 0 + 11] =11.  [13 *1 + 11] =24.  [13 * 2 + 11]=37.  [13 * 3 + 11] =50.....and so on until  [13 * 383  + 11] =4,990.

2 - N =156D  +  89, where D=0, 1, 2, 3......etc.

The smallest N = 89

5,000  -  89 =4,911

4,911 / 156 =31 - these are multiples of 156 between 89 and 5,000.

So, you have: 31 + 1 =32 integers between 1 and 5,000.

Follow exactly the same procedure as above.

Nov 14, 2020