Prime numbers

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

Find the highest power of 11 that will divide into 1000! exactly.

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?