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### Modulus Congruence Proof

```
Date: 04/18/2001 at 03:11:48
From: Ooi Chin Wah
Subject: Number theory

would be very grateful if you can help me again.

Prove 2^(3n+2)+21n = 4 mod (49)

Thanks.
```

```
Date: 04/18/2001 at 10:01:54
From: Doctor Rob
Subject: Re: Number theory

Thanks for writing back to Ask Dr. Math.

First let's verify the congruence modulo 7. We know 2^3 = 1 (mod 7).
That implies that:

2^(3*n+2) = 4*(2^3)^n = 4*1^n = 4 (mod 7)
2^(3*n+2) + 21*n = 4 (mod 7)

and the congruence holds modulo 7. Now we know that:

2^3 = 1 + 7
2^(3*n) = (1+7)^n

Expand using the Binomial Theorem, and reduce modulo 49 = 7^2. Then

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory
High School Number Theory

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