Combinatorics | Number Theory Answers

if a, b, c are any three integers such that (a,c)=1 and (b,c)=1, then show that (ab,c)=1.

Find all integral solutions of x^2 +1≅ 1(mod 5^3).

Find the smallest integer a>5 such that 4|a, 6|(a+1), 7| (a+2) and 8|(a+1)

Let a, b, c be integers such that gcd(a, b, c)= 1. Find gcd (a+b, b+c, c).

prove that gcd (n-1, n+1)= 1 or 2 for each n>=2 and gcd (2n-1, 2n+1)= 1 for each n>=7.

Let a, b, c be any three integers. Define gcd(a, b, c), the gcd od a, b, c as a positive integer d such that
(i) d|a, d|b, d|c and
(ii) if f|a, f|b, f|c then f|c
prove that gcd(a, b, c) =gcd(a, (b,c)) =gcd ((a,b),c) = gcd ((a,c),b)

When eggs in a basket are taken out 2, 3 , 4, 5 , 6at a time, there remain respectively 1, 2, 3, 4, 5 eggs; while the number comes even when they are taken out seven at a time. Find the smallest number of eggs in basket.

solve the linear congruence 17x≅9(mod 276), by using Chinese Remainder Theorem.

If a pizza costs £2.85 to produce and is sold to the guest for £9.95, what percentage gross profit is made from this dish?

Explain how would you find the empirical probability of rolling a 4 on a die?