Square numbers always have an even number of factors. True or false? N1.3 Core Starter Square numbers always have an even number of factors. Apart from 1, any whole number is either prime or can be made by combining multiples of just two different prime numbers. For example, 8 = 3 + 3 + 2. Add 1 to any multiple of 6. The result is always a prime number. All prime numbers greater than 11 can be written as the sum of two square numbers. For example, 13 = 32 + 22. Preamble For this activity pupils will need to be clear that a mere ‘true’ or ‘false’ is not sufficient and that they need to justify their answers. Possible content Number properties of integers and the associated vocabulary, trial and improvement, systematic working and recording. Resources None. Solution/Notes False – square numbers always have an odd number of factors. True – any even number by definition is made up of 2s. Any odd number can be made up of an even number plus 3 (2 and 3 are both prime). False – 6 + 1 = 7 (prime), 12 + 1 = 13 (prime), 18 + 1 = 19 (prime) but 24 + 1 = 25 (not prime). False – 19 cannot be written as the sum of two square numbers. Original Material © Cambridge University Press 2009 Original Material © Cambridge University Press 2009