A six-digit number starts with the digit \(1\). If this digit is relocated to the rightmost position, the number becomes \(3\) times bigger. What is the number?
Shmerlin managed to enter the cave and explore it. On his way back, he was once again stopped by Drago. He learns that the door out of the cave is locked again, this time with a more powerful lock. The key required to open it now includes four positive integers, which are no longer digits – they can be much larger. Shmerlin once again can choose four integer numbers: \(x, y, z\) and \(w\), and the dragon will tell him the value of \(A \times x + B \times y + C \times z + D \times w\), where \(A, B, C\) and \(D\) are the four secret integer numbers that open the lock. Because the lock is much more difficult to crack now, Drago agrees to let Shmerlin try twice. He can choose his four integer numbers and then, basing on what he learns from the dragon, choose again. Will he be able to leave the cave or is he doomed to stay inside forever?
Integer numbers \(a,b\) and \(c\) are such that the sum of digits of a number \(a+b\) is less than \(5\), the sum of digits of a number \(b+c\) is less than \(5\), the sum of digits of a number \(a+c\) is less than \(5\), but the sum of digits of a number \(a+b+c\) is greater than \(50\). Can you find such three numbers \(a,b\) and \(c\)?
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?