In a room, there are three-legged stools and four-legged chairs. When people sat down on all of these seats, there were 39 legs (human and stool/chair legs) in the room. How many stools are there in the room?
Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.
a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.
b) Will this statement remain true if instead of the difference we considered the total?
Two classes with the same number of students took a test. Having checked the test, the strict teacher Mr Jones said that he gave out 13 more twos than other marks (where the marks range from 2 to 5 and 5 is the highest). Was Mr Jones right?
Is the number \(10^{2002} + 8\) divisible by 9?
Will the quotient or the remainder change if a divided number and the divisor are increased by 3 times?
Try to get one billion \(1000000000\) by multiplying two whole numbers, in each of which there cannot be a single zero.
Are the sum and product odd or even for:
a) two even numbers?
b) two odd numbers?
c) an odd and an even number?
A group of numbers \(A_1, A_2, \dots , A_{100}\) is created by somehow re-arranging the numbers \(1, 2, \dots , 100\).
100 numbers are created as follows: \[B_1=A_1,\ B_2=A_1+A_2,\ B_3=A_1+A_2+A_3,\ \dots ,\ B_{100} = A_1+A_2+A_3\dots +A_{100}.\]
Prove that there will always be at least 11 different remainders when dividing the numbers \(B_1, B_2, \dots , B_{100}\) by 100.
Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.