Combinatorics | Number Theory Answers

Add two digits after 16 in such a way that the new 4-digit number is divisible by 3,4 and is the lowest one.Find out that new 4-digit number.

in how many ways can a committee of 5 be formed from agroup of 11 people consisting of 4 teachers and 7 students if
- there is no restriction in the selection?
- the committee must include exactly 2 teachers?

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.