Factors and multiples

problem
Phew I'm factored

Age

14 to 16

Challenge level

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.
problem
The public key

Age

16 to 18

Challenge level

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.
problem
Data chunks

Age

14 to 16

Challenge level

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and blue chunks, explore what sizes near to 31 can, or cannot, be exactly filled.
problem
There's always one isn't there

Age

14 to 16

Challenge level

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.
problem
Rarity

Age

16 to 18

Challenge level

Show that it is rare for a ratio of ratios to be rational.
problem
Expanding zeros

Age

11 to 14

Challenge level

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?
problem
Trailing zeros

Age

11 to 14

Challenge level

How many zeros does 50! have at the end?
problem
Factor list

Age

14 to 16

Challenge level

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
problem
Biscuit decorations

Age

5 to 7

Challenge level

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?
problem
Constant counting

Age

5 to 7

Challenge level

You can make a calculator count for you by any number you choose. You can count by ones to reach 24. You can count by twos to reach 24. What else can you count by to reach 24?