{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT6_042JF10_assn03

# MIT6_042JF10_assn03 - 6.042/18.062J Mathematics for...

This preview shows pages 1–2. Sign up to view the full content.

6.042/18.062J Mathematics for Computer Science September 21, 2010 Tom Leighton and Marten van Dijk Problem Set 3 Problem 1. [16 points] Warmup Exercises For the following parts, a correct numerical answer will only earn credit if accompanied by it’s derivation. Show your work. (a) [4 pts] Use the Pulverizer to find integers s and t such that 135 s + 59 t = gcd(135 , 59). (b) [4 pts] Use the previous part to find the inverse of 59 modulo 135 in the range { 1 , . . . , 134 } . (c) [4 pts] Use Euler’s theorem to find the inverse of 17 modulo 31 in the range { 1 , . . . , 30 } . (d) [4 pts] Find the remainder of 34 82248 divided by 83. ( Hint: Euler’s theorem. ) Problem 2. [16 points] Prove the following statements, assuming all numbers are positive integers. (a) [4 pts] If a | b , then c , a | bc (b) [4 pts] If a | b and a | c , then a | sb + tc . (c) [4 pts] c , a | b ca | cb (d) [4 pts] gcd( ka, kb ) = k gcd( a, b ) Problem 3. [20 points] In this problem, we will investigate numbers which are squares modulo a prime number p . (a) [5 pts] An integer n is a square modulo p if there exists another integer x such that n x 2 (mod p ). Prove

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

MIT6_042JF10_assn03 - 6.042/18.062J Mathematics for...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online