MAS111_09_assignment_02_hints

MAS111_09_assignment_02_hints - NANYANG TECHNOLOGICAL...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 2 HINTS 1. For any positive odd numbers m,n , prove that m 2 + n 2 is even, but not a multiple of 4. Hint : Let m = 2 k + 1 and n = 2 l + 1, and calculate m 2 + n 2 . 2. Prove that for any natural number n, 3 does not divide n 2 + 1. Hint : Consider the cases when n = 3 k + i for integer k and i = 0 , 1 , 2. 3. Prove that √ 5 + √ 7 is an irrational number. Hint : Prove it by contradiction. Suppose that √ 5+ √ 7 is a rational number then √ 5- √ 7 is also rational (why?). Now √ 5 = 1 2 (( √ 5 + √ 7) + ( √ 5- √ 7)) is again rational, which is not true (you may prove here that √ 5 cannot be rational by one more contradiction proof). 4. Prove that for any natural number n , 2 2 n- 1 is divisible by 3. Hint : Prove it by induction. Note that 2 2( k +1)- 1 = 2 2( k +1)- 2 2 k + 2 2 k- 1 = 3 · 2 2 k + 2 2 k- 1....
View Full Document

This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

Page1 / 3

MAS111_09_assignment_02_hints - NANYANG TECHNOLOGICAL...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online