Is \(100\times 99 \times ...\times 2\) divisible by \(2^{100}\)?
Prove the magic trick for the number \(1089
= 33^2\): if you take any \(3\)-digit number \(\overline{abc}\) with digits coming in
strictly descending order and subtract from it the number obtained by
reversing the digits of the original number \(\overline{abc} - \overline{cba}\) you get
another \(3\)-digit number, call it
\(\overline{xyz}\). Then, no matter
which number you started with, the sum \(\overline{xyz} + \overline{zyx} =
1089\).
Recall that a number \(\overline{abc}\)
is divisible by \(11\) if and only if
\(a-b+c\) also is.
Which of the following numbers are divisible by \(11\) and which are not? \[121,\, 143,\, 286, 235, \, 473,\, 798, \, 693,\, 576, \,748\] Can you write down and prove a divisibility rule which helps to determine if a three digit number is divisible by \(11\)?
Katie and Charlotte had \(4\) sheets of paper. They cut some of the sheets into \(4\) pieces. They then cut some of the newly obtained papersheets also into \(4\) pieces. They did this several more times, cutting a piece of paper into \(4\). In the end they counted the number of sheets. Could this number be \(2024\)?
The distance between two villages equals \(999\) kilometres. When you go from one village to the other, every kilometre you see a sign on the road, saying \(0 \mid 999, \, 1\mid 998, \, 2\mid 997, ..., 999\mid 0\). The signs show the distances to the two villages. Find the number of signs that contain only two different digits. For example, the sign \(0\mid999\) contains only two digits, namely \(0\) and \(9\), whereas the sign \(1\mid998\) contains three digits, namely \(1\), \(8\) and \(9\).
A monkey becomes happy when they eat three different fruits. What is the largest number of monkeys that can become happy with \(20\) pears, \(30\) bananas, \(40\) peaches and \(50\) tangerines?
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
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?