Math 250
Final Exam
1. (a) Let n be a 4-digit palindrome number. Prove that n is divisible by 11.
[5]
(b) Let m be a 5-digit palindrome number. What condition on m is sufficient to ensure that
m is divisible by 11?
✓ ◆
5
2. Let p be an odd prime di↵erent from 5. Prove that
= 1 if and only if p ⌘ 1, 3, 7, 9 mod 20.
p
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3. For what values of a does the congruence ax ⌘ 2 mod 13 have a solution?
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4. Let a, b, c be integers. Suppose that p is an odd prime which does not divide a.
(a) Show the number of solutions to
is given by 1 +
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b2
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ax2 + bx + c ⌘ 0 mod p
◆
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✓ ◆
4ac
·
0
where
is the Legendre symbol and
=0
p
p
p
(b) Find all the solutions to 2x2 + 9x + 10 ⌘ 1 mod 11.
5. (a) Conjecture a relationship between (n) and (n2 ).
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(b) Prove your conjecture is true.
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6. (a) Find all the solutions to the Diophantine equation y 2 = x5 + 1 modulo 7.
(b) Find all the solutions to the Diophantine equation
y2
=
x5
+ 1 modulo 11.
(c) Let p be a prime with the property that p 6⌘ 1 mod 5. Prove that the Diophantine equation
y 2 = x5 + 1 has exactly p solutions modulo p.
p
p
7. Let Z[ 7] = {a + b 7 | a, b in Z}.
p
p
(a) Show that Z[ 7] is a ring. Inpother words, show that if ↵ and are in Z[ 7] , then ↵ + ,
↵
, and ↵ are also in Z[ 7].
p
p
(b) Define the norm
of an element in Z[ 7] as N (a + b 7) = a2 7b2 . Show that for any ↵
p
and in Z[ 7], N (↵ ) = N (↵)N ( ).
p
(c) Factor 31 in Z[ 7].
✓ ◆
·
8. Let
denote the Legendre symbol modulo p. Suppose that p 2 is a prime number such
p
that p ⌘ 1 mod 8. Compute the following:
✓
◆
p+1
(a)
p
✓
◆
p 1
(b)
p
✓
◆
(p 1)(p + 2)
(c)
p
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