Question

Prove: If n ≡ 3 (mod 8) and n = a^{2} + b^{2} +
c^{2} + d^{2}, then exactly one of a, b, c, d is
even. (Hint: What can each square be modulo 8?)

Answer #1

Prove: If n≡3 (mod 8) and n=a^2+b^2+c^2+d^2, then exactly one of
a, b, c, d is even. (Hint: What can each square be modulo 8?)

Prove the statement: For all integers a, b,and c, if
a2 + b2 = c2, then a or b is
even.

Let a, b ∈Z and n ∈N.
(a) True/False: If a2 ≡ b2 (mod n), then a
≡ b (mod n).
(b) True/False: If a ≡ b (mod p) and a ≡ b (mod q) for distinct
primes p and q, then a ≡ b (mod pq).
Justify your answers properly.

Prove that if n is a positive integer greater than 1,
then n! + 1 is odd
Prove that if a, b, c are integers such that a2 + b2 =
c2, then at least one of a, b, or c is even.

There are two boxes, A1×B1×C1 and A2×B2×C2 are size of boxes.
Define if it is possible to totally cover one box in the another.
(Hint: 1x1x1 can be covered by 2x1x1 box)
Input format
A1, B1, C1, A2, B2, C2.
Output format
The program should bring out one of the following lines:
Boxes are equal, if the boxes are the same,
the first box is smaller than the second one, if the first box
can be put in the second,...

a) Prove: If n is the square of some integer, then n /≡ 3 (mod
4). (/≡ means not congruent to)
b) Prove: No integer in the sequence 11, 111, 1111, 11111,
111111, . . . is the square of an integer.

9e) fix n ∈ ℕ. Prove congruence modulo n is an equivalence
relation on ℤ. How many equivalence classes does it have?
9f) fix n ∈ ℕ. Prove that if a ≡ b mod n and c ≡ d mod n then a
+ c ≡b + d mod n.
9g) fix n ∈ ℕ.Prove that if a ≡ b mod n and c ≡ d mod n then ac
≡bd mod n.

Prove: Proposition 11.13. Congruence modulo n is an equivalence
relation on Z :
(1) For every a ∈ Z, a = a mod n.
(2) If a = b mod n then b = a mod n.
(3) If a = b mod n and b = c mod n, then a = c mod n

The smallest positive solution of the congruence ax ≡ 0( mod n)
is called the additive order of a modulo n. Find the additive
orders of each of the following elements by solving the appropriate
congruences: (abstract algebra)
(a) 20 modulo 28
(b) 8 modulo 15
(c) 7 modulo 11
(d) 9 modulo 15

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

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