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Show that if n eZ is a perfect square such that 3 n, then 9|n.

Show that if n eZ is a perfect square such that 3 n, then 9|n.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q3 maybe division theorem
00:07 ul 5G Вack Homework8.pdf Math 109 Homework 8 Due: December 4, 2021 at 11:59 PM 1. Suppose that a, b e Z with b > 0. By the division theorem, there exist q, r eZ such that a = bq +r and 0<r < b. Show that b divides a if and only if r = 0. 2. Use the division theorem to show that if n e Z is a perfect square (i.e. n = m² for some m e Z), then there exists qe Z such that n = 3q or n = 3g + 1. Deduce that 29,537 is not a perfect square. 3. Show that if n e Z is a perfect square such that 3|n, then 9|n. 4. Let a, b be nonzero integers, and suppose that a = bq + r for some q,r e Z. Show that ged(a, b) = gcd(b, r) 5. Suppose a1, a2, b1, bz E Z4 with a,b2 = azbı. Show that if aj and by are coprime and az and b2 are coprime (i.e. ged(a1, b1) = gcd(a2, b2) = 1), then a1 = az and bị = b2. 6. Find all solutions to the equation 165m + 252n = 15. Show your work. 7. Find all solutions to the equation 336m + 238n = 5558. Show your work. 119 Dashboard Calendar To Do Notifications Inbox
Transcribed Image Text:00:07 ul 5G Вack Homework8.pdf Math 109 Homework 8 Due: December 4, 2021 at 11:59 PM 1. Suppose that a, b e Z with b > 0. By the division theorem, there exist q, r eZ such that a = bq +r and 0<r < b. Show that b divides a if and only if r = 0. 2. Use the division theorem to show that if n e Z is a perfect square (i.e. n = m² for some m e Z), then there exists qe Z such that n = 3q or n = 3g + 1. Deduce that 29,537 is not a perfect square. 3. Show that if n e Z is a perfect square such that 3|n, then 9|n. 4. Let a, b be nonzero integers, and suppose that a = bq + r for some q,r e Z. Show that ged(a, b) = gcd(b, r) 5. Suppose a1, a2, b1, bz E Z4 with a,b2 = azbı. Show that if aj and by are coprime and az and b2 are coprime (i.e. ged(a1, b1) = gcd(a2, b2) = 1), then a1 = az and bị = b2. 6. Find all solutions to the equation 165m + 252n = 15. Show your work. 7. Find all solutions to the equation 336m + 238n = 5558. Show your work. 119 Dashboard Calendar To Do Notifications Inbox
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