We call a \(10\)-digit number interesting if it is divisible by \(11111\), and all its digits are different. How many interesting numbers does there exist?
Note that a number \(k = a_0 + 10a_1 + \dots +10^9 a_9\) is divisible by \(11111\) if and only if a number \(m = (a_0+a_5) +10(a_1+a_6) + \dots + 10^4 (a_4+a_9)\) is also divisible by \(11111\). This is because \(100000=1+9 \times 11111\) and we subtract \(99999 (a_5 + 10a_6 + 100a_7 + 1000a_8 +10000a_9)\) from the original number.
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?
The number \(b^2\) is divisible by \(8\). Show that it must be divisible by \(16\).
Find a number which:
a) It is divisible by \(4\) and by \(6\), is has a total of 3 prime factors, which may be repeated.
b) It is divisible by \(6, 9\) and \(4\), but not divisible by \(27\). It has \(4\) prime factors in total, which may be repeated.
c) It is divisible by \(5\) and has exactly \(3\) positive divisors.
a) The number \(a\) is even. Should \(3a\) definitely also be even?
b) The number \(5c\) is divisible by \(3\). Is it true that \(c\) is definitely divisible by \(3\)?
c) The product \(a \times b\) is divisible by \(7\). Is it true that one of these numbers is divisible by \(7\)?
d) The product \(c \times d\) is divisible by \(26\). Is it true that one of these numbers is divisible by \(26\)?
a) The number \(a^2\) is divisible by \(11\). Is \(a^2\) necessarily also divisible by \(121\)?
b) The number \(b^2\) is divisible by \(12\). Is \(b^2\) necessarily also divisible by \(144\)?
What is the smallest integer \(n\) such that \(n\times (n-1)\times (n-2) ... \times 2\) is divisible by \(990\)?
Jack believes that he can place \(99\) integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?
a) Prove that a number is divisible by \(8\) if and only if the number formed by its laast three digits is divisible by \(8\).
b) Can you find an analogous rule for \(16\)? What about \(32\)?