Highest common factor and lowest common multiple

Exact product

\(\displaystyle 6\times4=24\)

Which statement is correct?

Common means shared by both

A common factor divides both numbers exactly. The highest common factor, or HCF, is the largest positive factor they share

The shared factors are one, two, four and eight. Eight is the largest

One divides every positive whole number, so every pair has at least one common factor

Change the candidate divisor through every setting. A common factor leaves zero remainder in both rows. No positive common factor can exceed eighteen, the smaller number

What is the largest setting with zero remainder in both rows?

Use the smaller shared prime counts

A common factor must divide both numbers, so it cannot use more copies of a prime than either number has

Both numbers have three copies of two. Three is missing from forty, and five is missing from twenty-four, so neither can be included

Multiplying eight by another prime would stop it dividing at least one of the numbers. That is why this common factor is the highest

Find the HCF from the original numbers

Find the HCF

Divide \(20\) by prime factors until the quotient is one

Divide \(30\) by prime factors until the quotient is one

A common factor cannot use more copies of a prime than either number has. Use the smaller count of each prime

\(2\) has count 2 in \(20\) and count 1 in \(30\). Use 1 copy

\(3\) has count 0 in \(20\) and count 1 in \(30\). It is not shared, so omit it

\(5\) has count 1 in \(20\) and count 1 in \(30\). Use 1 copy

Check that the result divides both original numbers

Including another prime factor would exceed the available count in at least one number, so this common factor is the highest

Keep only the shared copies

Find the first positive common multiple

A common multiple is a multiple of both numbers. The lowest common multiple, or LCM, is the smallest positive one

These lists start with one copy of twelve and one copy of eighteen. Their first shared value is thirty-six

Zero is a multiple of both numbers, but LCM asks for a positive common multiple. Zero is not the answer

Use the larger required prime counts

A common multiple must contain enough prime factors to include either original number

Three copies of two are enough for both numbers. Also include one three for twenty-four and one five for forty

The three copies of two can serve both numbers. They are not needed twice. Removing any selected prime factor would leave too few copies for at least one number

Build the lowest common multiple

Find the LCM

Divide \(18\) by prime factors until the quotient is one

Divide \(30\) by prime factors until the quotient is one

A common multiple must contain enough copies of each prime for both numbers. Use the larger count of each prime

\(2\) has count 1 in \(18\) and count 1 in \(30\). Use 1 copy

\(3\) has count 2 in \(18\) and count 1 in \(30\). Use 2 copies

\(5\) has count 0 in \(18\) and count 1 in \(30\). Use 1 copy

Check that the result is a multiple of both original numbers

Removing a selected prime factor would leave too few copies for at least one number, so this positive common multiple is the lowest

Include every required prime

First number

\(\displaystyle 12=2^{2}\times3\)

Second number

\(\displaystyle 18=2\times3^{2}\)

Which pair of results is correct?

No shared prime factors

Eight and fifteen share no prime factors. Such numbers are called coprime

Their HCF is one, not zero. One still divides both

Their LCM needs all the primes from both numbers, because none is shared

For a coprime pair, multiplying the two numbers gives the LCM. This shortcut does not work for every pair

A common factor still exists

Include the unshared prime factors

Find the LCM

Divide \(8\) by prime factors until the quotient is one

Divide \(15\) by prime factors until the quotient is one

A common multiple must contain enough copies of each prime for both numbers. Use the larger count of each prime

\(2\) has count 3 in \(8\) and count 0 in \(15\). Use 3 copies

\(3\) has count 0 in \(8\) and count 1 in \(15\). Use 1 copy

\(5\) has count 0 in \(8\) and count 1 in \(15\). Use 1 copy

Check that the result is a multiple of both original numbers

Removing a selected prime factor would leave too few copies for at least one number, so this positive common multiple is the lowest

When one number divides the other

Every prime factor of twelve is already available in thirty-six. The smaller counts recover twelve, and the larger counts recover thirty-six

The smaller number is the HCF and the larger is the LCM only when one divides the other exactly

One contributes no prime factors. With one and twelve, the HCF is one and the LCM is twelve

Check a pair with an exact division

Find the LCM of that pair

Choose HCF for equal groups

Eighteen buttons and thirty beads are shared into as many identical sets as possible, using every item. The number of sets must divide both counts exactly

Make six sets, each with three buttons and five beads. Six is the greatest common factor, so no larger number of identical sets can use everything without leftovers

Make as many equal packs as possible

Choose LCM for the next shared event

Two timers sound together now. One repeats every six minutes and the other every ten minutes. Find when they next sound together

The elapsed time must be a positive multiple of both intervals. Choose the first such time

They next sound together after thirty minutes. Zero is the starting time, not the next event

Find the next simultaneous event

Check the direction and the prime counts

To find the HCF, use only shared prime factors and the smaller count of each. Check that the answer divides both original numbers

To find the LCM, use every required prime and the larger count of each. Check that the answer is a positive multiple of both original numbers

Equal grouping asks for a common factor. Repeating events meeting again ask for a positive common multiple. The words greatest and next decide which shared value is needed

A pair with no shared primes has HCF one. Its LCM is the product. If one number divides the other, the smaller is the HCF and the larger is the LCM