Factors, Multiples, LCM and HCF

A factor of a number divides into it with nothing left over, and every factor has a partner: the number you multiply it by to get back. So find factors in pairs, from the outside in. For twenty-four: one times twenty-four, two times twelve, three times eight, four times six. Five does not go. Six would pair with four, which is already there — that is the sign you are finished. Eight factors, four pairs. Working in pairs means you never miss one, and you always know when to stop.

Multiples are the times table: the multiples of three are three, six, nine, and so on forever. When two numbers each have a times table, the first number that appears in both is their lowest common multiple. For three and eight it is twenty-four, and every later shared number is a multiple of twenty-four. The quick way is to list the multiples of the larger number and stop at the first one the smaller number divides: for six and eight, eight, sixteen, twenty-four — and six goes into twenty-four. Multiplying the two numbers together gives a common multiple, but usually not the lowest one. Buses every twelve minutes and trains every eighteen leave together every thirty-six minutes, not every two hundred and sixteen.

The highest common factor is the other direction: instead of the smallest number both go into, it is the largest number that goes into both. List the factors of each in pairs, ring the ones that appear in both lists, and take the biggest. Eighteen and twenty-four share one, two, three and six, so their highest common factor is six. It is the biggest equal group both numbers can be split into: eighteen is three sixes and twenty-four is four sixes. Keep the two ideas apart: the HCF of twelve and eighteen is six, but their LCM is thirty-six.

Here is the selective test's favourite use of all this. How many multiples of three from one to one hundred are not multiples of eight? First count the multiples of three: one hundred divided by three is thirty-three remainder one, so there are thirty-three of them. Then take away the ones that are also multiples of eight. A number that is a multiple of both three and eight is a multiple of their LCM, twenty-four, and up to a hundred there are four of those: twenty-four, forty-eight, seventy-two, ninety-six. Thirty-three minus four is twenty-nine. Divide to count, use the LCM for both, subtract for not.

The last shape is a puzzle: numbers described by conditions, and a question about the extremes. Kim's number is a multiple of five and less than fifty, so the largest it could be is forty-five, because fifty is not less than fifty. Jamie's number is even, a multiple of seven and more than zero: the multiples of seven are seven, fourteen, twenty-one, and the even ones start at fourteen, so the smallest is fourteen. The difference between the largest Kim and the smallest Jamie is forty-five minus fourteen, thirty-one. Each condition trims the list; write the list, apply the next condition, and read the boundaries carefully. Quiz and exercises now.