Can you come up with a divisibility rule for \(5^n\), where \(n=1\), \(2\), \(3\), . . .? Prove that the rule works.
Show that for each \(n=1\), \(2\), \(3\), . . ., we have \(n<2^n\).
Show that \(n^2+n+1\) is not divisible by \(5\) for any natural number \(n\).
Given a natural number \(n\), find a formula for the number of \(k\) less than \(n\) such that \(k\) is coprime to \(n\). Prove that the formula works.
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).
Ms Jones vacuums her car every 2 days, she washes her car every 7 days and polishes it every 52 days. The last time she did all three types of cleaning on one day was on the 13th of March last year. What time will she do it again?
The numbers \(a\) and \(b\) are integers and \(a>b\). Show that the gcd of \(a\) and \(b\) is equal to the gcd of \(b\) and \(a-b\).
A brave witch is out there hunting monsters for coins. She noticed that every 5th monster she encounters has wings, every 16th has a fiery breath, every 6th has fangs and every 14th has a pile of treasure. Now, the only monster with wings, fiery breath, fangs and a pile of treasure is a dragon and witches don’t hunt dragons. Suppose that the witch has just met a dragon and that every dragon has wings, firey breath, fangs and a pile of treasure. How many monsters will she have to hunt to meet another one?
Let \(a = 8 \times 9^2 \times 31^2 \times 7\) and \(b= 7^2 \times 2^3 \times 3^6 \times 23^2\). Find their greatest common divisor and least common multiple.
Let \(n\) be a nonnegative integer. What is the gcd of \(12n+9\) and \(9n+6\)?