Download Problem set 2

MATH 210
Problem Set 2
Due: January 22, 2016
Exercises
1. Find all solutions to
(b) 121x − 88y = 572.
(a) 15x + 27y = 1,
2. Find a complete set of mutually incongruent solutions of
(i) 8x ≡ 10
(ii) 15x ≡ 715
(mod 30),
(mod 2187).
3. Show that there are infinitely many primes of the form 6x + 5, e.g., 5, 11, 17, 23, 29, 41, . . .
Problems
4. The textbook covers divisibility criteria for some small integers in Proposition 3.2, but has left off
everyone’s favourite (prime) number: 37.
(a) Show that if a three digit number
abc = 100a + 10b + c
is divisible by 37, then so are bca and cab.
(b) Show that if the sum of two three digit numbers abc and def is divisible by 37, then their
concatenation abcdef (a six digit number) is also divisible by 37.
5. (a) Consider the polynomial
p(x) = a0 + a1 x + a2 x2 + · · · + an xn ,
where a0 , a1 , . . . , an ∈ Z. Show that if p(xi ) = 7 for 4 distinct integers x0 , x1 , x2 , x3 , then p(z) 6= 14
for any z ∈ Z.
(b) Let x, y ∈ Z, and let a, n be non-zero integers. Show that ax ≡ ay (mod n) if and only if
n
x≡y
mod
.
gcd(a, n)
Food for thought
We have seen many ways that modular arithmetic captures integer arithmetic quite nicely, but it turns out
the modular side of things cannot see everything integral.
For example, there are (Diophantine) equations that have solutions modulo n for every n > 1 and yet no
integral solutions. Find one.
Practice problems
Shifrin section 1.3: 2, 3, 5, 12, 13, 15, 16, 20, 34.
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