Question

2. (20 points) Given $a$, $b$, $c$, $d$, $n$ are integers with $n > 1$, assume $a \equiv c \pmod{n}$ and $b \equiv d \pmod{n}$. Prove each of the following statements. (a) $a^2 \equiv b^2 \pmod{n}$ (b) $a^m \equiv b^m \pmod{n}$ for every integer $m \ge 1$ using induction on $m$. (c) $(a+c) \equiv (b+d) \pmod{n}$ (d) $(a^2 + c) \equiv (b^2 + d) \pmod{n}$

          2. (20 points) Given $a$, $b$, $c$, $d$, $n$ are integers with $n > 1$, assume $a \equiv c \pmod{n}$ and $b \equiv d \pmod{n}$.
Prove each of the following statements.
(a) $a^2 \equiv b^2 \pmod{n}$
(b) $a^m \equiv b^m \pmod{n}$ for every integer $m \ge 1$ using induction on $m$.
(c) $(a+c) \equiv (b+d) \pmod{n}$
(d) $(a^2 + c) \equiv (b^2 + d) \pmod{n}$

2. (20 points) Given a, b, c, d, n are integers with n > 1, assume a ≡ c n and b ≡ d n.
Prove each of the following statements.
(a) a^2 ≡ b^2 n
(b) a^m ≡ b^m n for every integer m ≥ 1 using induction on m.
(c) (a+c) ≡ (b+d) n
(d) (a^2 + c) ≡ (b^2 + d) n

Added by Carl C.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Adi S

11/15/2022

Step 1

This means that a and c have the same remainder when divided by n, and b and d have the same remainder when divided by n. Show more…

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2. (20 points) Given a, b,c, d, n are integers with n > 1, assume a = c (mod n) and b = d (mod n) Prove each of the following statements. (a) a2 = b2 (mod n) (b) am = bm (mod n) for every integer m 1 using induction on m. (c) (a+c) =(b+d) (mod n) (d) (a2 + c) = (b2+ d) (mod n)