Problems

Find three different natural numbers, larger than \(100\) such that each of them is divisible by the difference of the other two numbers? The values of differences also have to be different from each other.

There are four numbers written in a row. The first number is \(100\). It is known that if we divide the first number by the second number we will get a prime number as a result, if we the second number by the third number we will get a prime number, and if we divide the third number by the fourth number we will also get a prime number. Can all the resulting prime numbers be distinct?

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 \(a\) has a prime factorization \(2^3 \times 3^2 \times 7^2 \times 11\). Is it divisible by \(54\)? Is it divisible by \(154\)?

Two numbers are given in terms of their prime factorizations: \(a= 2^3 \times 3^2 \times 5 \times 11^2 \times 17^2\) and \(b = 2 \times 5^3 \times 7^2 \times 11 \times 13\).

a) What is the greatest common divisor \(\mathrm{gcd}(a,b)\) of these numbers?

b) What is their least common multiple \(\mathrm{lcm}(a,b)\)?

c) Write down the prime factorization of \(\mathrm{gcd}(a,b) \times \mathrm{lcm}(a,b)\). Then write the prime factorization of \(a \times b\). What do you notice?

The meeting of the secret agents took place in the green house.

Considering the numbers in the windows of the green house, what should be drawn in the empty frame?

Find one way to encrypt letters of Latin alphabet as sequences of \(0\)s and \(1\)s, each letter corresponds to a sequence of five symbols.

Pinoccio keeps his Golden Key in the safe that is locked with a numerical password. For secure storage of the Key he replaced some digits in the password by letters (in such a way that different letters substitute different digits). After replacement Pinoccio got the password \(QUANTISED17\). Honest John found out that:
• the number \(QUANTISED\) is divisible by all integers less than 17, and
• the difference \(QUA-NTI\) is divisible by \(7\).
Could he find the password?

Using the representation of Latin alphabet as sequences of \(0\)s and \(1\)s five symbols long, encrypt your first and last name.

Decipher the quote from Philip Pullmans "His Dark Materials":
Erh csy wlepp orsa xli xvyxl, erh xli xvyxl wlepp qeoi csy jvii.
The same letters correspond to the same in the phrase, different letters correspond to different. We know that no original letters stayed in place, meaning that in places of e,r,h there was surely something else.