Practice Problems for Chinese Remainder Theorem
1.
Can the sequence of positive integers 1, 2, 3, … be rearranged in a way that the
sum of the first k terms is divisible by k for all positive integers k?
2.
Prove that for every positive integer n, there are n consecutive positive integer
such that each of them is square-free.
3.
Prove that for every positive integer n, there are n consecutive positive integer
such that none of them is square-free.
4.
Let m, n be positive integers satisfying:
for any positive integer k, gcd(11k  1, m)  gcd(11k  1, n) .
Prove that there exist a integer l such that m  11l n .
5.
Are there exist 21 consecutive positive integers such that each of them is
divisible by at least one of the numbers 2, 3, 5, 7, 11, 13?
6.
Let f (n) be the least positive integer such that
f ( n)
k
is divisible by n.
k 1
Prove that f (n)  2n  1 if and only if n  2 m (m is nonnegative integer).
7.
Can we find a set S containing exactly 1990 positive integers satisfying:
(i.) The elements in S are relatively prime in pairs,
(ii.) The sum of any k ( 2) elements in S is composite.