Is it possible to divide the numbers 1, 2, 3, ..., 100 into pairs of one odd and one even number, such that in every pair except one the even number is greater than the odd number
Back in the days when a young mathematician was even younger he could only draw digits “4” and “7”. While looking through the old notes his mother found one piece of paper on which he wrote the numbers with digit sums equal to 18, 22 and 26. Which numbers could be written on this piece of paper?
a) It seems that the young mathematician was making progress quite fast. On the back side of that piece of paper there are numbers with digits adding up to all natural numbers from 18 to 33. And yet all of them consist of only digits “4” and “7”. Make your own list of that kind.
(b) Is it true that any natural number greater than 17 can be equal to the digit sum of some number written with digits “4” and “7”?
(c) Now let’s try the same question for digits “5” and “8”. What values can you get if you consider the sum of the digits of a number written with the help of digits “5” and “8”?
Is it true that if a natural number is divisible by \(4\) and by \(6\), then it must be divisible by \(4\times6=24\)?
A natural number is divisible by \(5\) and by \(7\). Should it be divisible by \(35\)?
The number \(A\) is not divisible by 3. Is it possible that the number \(2A\) is divisible by 3?
Lisa knows that \(A\) is an even number. But she is not sure if \(3A\) is divisible by \(6\). What do you think?
George divided number \(a\) by number \(b\) with the remainder \(d\) and the quotient \(c\). How will the remainder and the quotient change if the dividend and the divisor are increased by a factor of 3?
Let us introduce the notation – we denote the product of all natural numbers from 1 to \(n\) by \(n!\). For example, \(5!=1\times2\times3\times4\times5=120\).
a) Prove that the product of any three consecutive natural numbers is divisible by \(3!=6\).
b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?
Can a sum of three different natural numbers be divisible by each of those numbers?