The monkey becomes happy when they eat three different fruits. What is the largest number of monkeys that can get happy with \(20\) pears, \(30\) bananas, \(40\) peaches, and \(50\) tangerines?
The letters \(A\), \(R\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.
Find \(ART\), given that \(ARTS+STAR=10,T31\).
Split the numbers from \(1\) to \(9\) into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into \(124\), \(356\) and \(789\), then the triplet \(124\) is correct, since \(1+2+4=7\) is prime. But the other two triples are incorrect, since \(3+5+6=14\) and \(7+8+9=24\) are not prime.
Let \(p\), \(q\) and \(r\) be distinct primes at least \(5\). Can \(p^2+q^2+r^2\) be prime? If yes, then give an example. If no, then prove it.
Is there a divisibility rule for \(2^n\), where \(n = 1\), \(2\), \(3\), . . .? If so, then explain why the rule works.
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 natural number \(n\), find a formula for the number of \(k\) such that \(k\) is coprime to \(n\). Prove the formula works.