Homework 10 Solution

Homework 10 Solution - MATH 74 HOMEWORK 10 (DUE WEDNESDAY...

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

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.

Unformatted text preview: MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28) 1(a). We first prove that if 5 | a , then 5 | a 2 . Suppose 5 | a . This means that there is q ∈ Z with a = 5 q . Squaring both sides we see a 2 = 25 q 2 = 5( q 2 ), so that 5 | a 2 . We now prove that if 5 | a 2 , then 5 | a . Suppose that 5 | a 2 . From the division theorem we know there are q ∈ Z and 0 ≤ r < 5 satisfying a = 5 q + r . If 5 6 | a then r 6 = 0, so that r ∈ { 1 , 2 , 3 , 4 } . But (5 q + 1) 2 = 25 q 2 + 10 q + 1 = 5(5 q 2 + 2 q ) + 1 (5 q + 2) 2 = 25 q 2 + 20 q + 4 = 5(5 q 2 + 4 q ) + 4 (5 q + 3) 2 = 25 q 2 + 30 q + 9 = 5(5 q 2 + 6 q + 1) + 4 (5 q + 4) 2 = 25 q 2 + 40 q + 16 = 5(5 q 2 + 8 q + 3) + 1 , so no matter what we see that 5 6 | a 2 , a contradiction. 1(b). The statement of 1(a) is false if 5 is replaced with 4; e.g. 4 | 2 2 but 4 6 | 2. 1(c). Numbers like 6 , 7 , 11 , 15 are like 5. Numbers like 9 , 12 , 18 , 25 are like 4. 2. We first prove that if 9 | a 2 then 3 | a 2 . To see this, note that if 9 | a 2 there is q ∈ Z with a 2 = 9 q . But then a 2 = 3(3 q ), showing that 3 | a 2 . We now show that if 3 | a 2 , then 9 | a 2 . Suppose 3 | a 2 . By the division theorem we have q ∈ Z and 0 ≤ r < 3 with a = 3 q + r . We cannot have r = 1 or r = 2, because then we would have either a 2 = (3 q + 1) 2 = 9 q 2 + 6 q + 1 = 3(3 q 2 + 2 q ) + 1 or a 2 = (3 q + 2) 2 = 9 q 2 + 12 q + 4 = 3(3 q 2 + 4 q + 1) + 1; in either case we’d conclude that 3 6 | a 2 , a contradiction. So, a contradiction....
View Full Document

This note was uploaded on 02/10/2010 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.

Page1 / 3

Homework 10 Solution - MATH 74 HOMEWORK 10 (DUE WEDNESDAY...

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

View Full Document
Ask a homework question - tutors are online