1. Number

Middle Years

Lesson

When we compare two numbers, we often want to know what their highest common factor and lowest common multiple are. These numbers help us understand how they are related, and are often the answer to problems involving them.

A number is a common factor of two other numbers if it divides both numbers without remainder. Below is a table with all the factors of $24$24 and $54$54, with their common factors listed in the last row:

Factors of $24$24 and $54$54 | |
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Factors of $24$24 | $1,2,3,4,6,8,12,24$1,2,3,4,6,8,12,24 |

Factors of $54$54 | $1,2,3,6,9,18,27,54$1,2,3,6,9,18,27,54 |

Common factors | $1,2,3,6$1,2,3,6 |

The common factors tell us how we can break up both numbers equally.

For example, $24$24 students and $54$54 books can be divided into:

- $2$2 equal groups with $12$12 students and $27$27 books each
- $3$3 equal groups with $8$8 students and $18$18 books each
- $6$6 equal groups with $4$4 students and $9$9 books each

The largest number in the list of common factors, $6$6, is called the highest common factor (HCF) of $24$24 and $54$54.

To find the HCF of two numbers we can always create a list of factors, like we did above. But there is a faster way using factor trees. To find the highest common factor of $126$126 and $294$294, we start by drawing their factor trees:

This gives us their prime factorisations in expanded form:

$126=2\times3\times3\times7$126=2×3×3×7

$294=2\times3\times7\times7$294=2×3×7×7

We then find what factors appear in both factorisations - they have one $2$2, one $3$3, and one $7$7 in common. The highest common factor is the product of the common prime factors, $2\times3\times7=42$2×3×7=42.

When we multiply two numbers together, these two numbers are always factors of the new number. Another way to look at it is to say the new number is a common multiple of both of the original ones.

Below is a table with the first few multiples of $12$12 and $18$18, with their common multiples listed in the last row:

Multiples of $12$12 and $18$18 | |
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Multiples of $12$12 | $12,24,36,48,60,72,84,96,108,\ldots$12,24,36,48,60,72,84,96,108,… |

Multiples of $18$18 | $18,36,54,72,90,108,126,144,\ldots$18,36,54,72,90,108,126,144,… |

Common multiples | $36,72,108,\ldots$36,72,108,… |

The common multiples are the numbers that can be broken up into both the original numbers equally.

- $36$36 can be broken up into $3$3 groups of $12$12, and $2$2 groups of $18$18
- $72$72 can be broken up into $6$6 groups of $12$12, and $4$4 groups of $18$18
- $108$108 can be broken up into $9$9 groups of $12$12, and $6$6 groups of $18$18

The smallest number in the list of common multiples, $36$36, is called the lowest common multiple (LCM) of $12$12 and $18$18.

To find the LCM of two numbers we can always create a list of multiples, like we did above. But just like for HCF, there is a faster way using factor trees. To find the lowest common multiple of $126$126 and $294$294, we look again at their factor trees:

We can instead jump straight to the prime factorisations in expanded form, since we found that already:

$126=2\times3\times3\times7$126=2×3×3×7

$294=2\times3\times7\times7$294=2×3×7×7

Now we create the number by multiplying one of the numbers by any prime factors it is missing from the other one.

There is only one $2$2 in each, there are two $3$3s in $126$126 (and only one in $294$294), but there are two $7$7s in $294$294.

If we multiply $126$126 by $7$7, the missing factor, we will find the lowest common multiple: $126\times7=882$126×7=882.

We would get the same result if we started with $294$294 - it has more than enough $7$7s, but it is missing the second $3$3 that $126$126 has. The multiplication $294\times3=882$294×3=882 gives us the same answer.

HCF and LCM

The highest common factor (HCF) of two numbers is the largest factor of both numbers. It is the product of the common prime factors between them.

The lowest common multiple (LCM) of two numbers is the smallest multiple of both numbers. Multiply one number by the prime factors it is missing from the other.

Consider the following prime factorisations:

$180=2\times2\times3\times3\times5$180=2×2×3×3×5

$600=2\times2\times2\times3\times5\times5$600=2×2×2×3×5×5

Find the highest common factor of $180$180 and $600$600.

Find the highest common factor of $150$150 and $560$560.

Consider the following prime factorisations:

$2200=2\times2\times2\times5\times5\times11$2200=2×2×2×5×5×11

$2750=2\times5\times5\times5\times11$2750=2×5×5×5×11

Find the lowest common multiple of $2200$2200 and $2750$2750.