If your year 11's / 5th formers are keeping you busy (as they do this time of year!) maybe try some of these questions with your other classes this month.

1. Take any two digit number. Reverse the digits (so 14 becomes 41 etc.). Now add the original number to the reversed number. Is the answer divisible by 11? Does this work for every two digit number? Test it then prove why it works.

2. Can you write the numbers 1 – 17 in a different order so that every pair of adjacent numbers adds up to make a square number?

3. a) What is 1 + 2 + 3 + … + 98 + 99 + 100

b) What is 1 + 2 + 3 + … + (n - 2) + (n – 1) + n?

4. List the first ten numbers with exactly three factors. How could you describe these numbers in general? Try to be as precise as possible.

5. A ‘perfect’ number is one whose factors (excluding itself) add up to make it. 6 is a perfect number since 1 + 2 + 3 = 6 (1, 2 and 3 are the factors of 6 if we exclude 6 itself). What is the next perfect number? The one after that? (it’s very big, but less than 1000!!!)

6. What is the next number in the following sequence…

10, 9, 60, 90, 70, 66, …

7. ‘Every integer greater than 2 can be expressed as the sum of two primes’. Prove it!

1. Take any two digit number. Reverse the digits (so 14 becomes 41 etc.). Now add the original number to the reversed number. Is the answer divisible by 11? Does this work for every two digit number? Test it then prove why it works.

2. Can you write the numbers 1 – 17 in a different order so that every pair of adjacent numbers adds up to make a square number?

3. a) What is 1 + 2 + 3 + … + 98 + 99 + 100

b) What is 1 + 2 + 3 + … + (n - 2) + (n – 1) + n?

4. List the first ten numbers with exactly three factors. How could you describe these numbers in general? Try to be as precise as possible.

5. A ‘perfect’ number is one whose factors (excluding itself) add up to make it. 6 is a perfect number since 1 + 2 + 3 = 6 (1, 2 and 3 are the factors of 6 if we exclude 6 itself). What is the next perfect number? The one after that? (it’s very big, but less than 1000!!!)

6. What is the next number in the following sequence…

10, 9, 60, 90, 70, 66, …

7. ‘Every integer greater than 2 can be expressed as the sum of two primes’. Prove it!