A math circle student Emilio wrote a computer program for his house robot, Basil. Starting from 1, Basil should keep writing bigger and bigger numbers formed by 1s: 1, 11, 111, etc. The program terminates when Basil writes a number that is a multiple of 19. Prove that the program will terminate in fewer than 20 steps.
We know that the product \(c \times d\) is divisible by a prime \(p\). Show that either \(c\) or \(d\) must be divisible by \(p\).
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.
The number \(a\) has a prime factorization \(2^3 \times 3^2 \times 7^2 \times 11\). Is it divisible by \(54\)? Is it divisible by \(154\)?
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 \(32\)?