Adding Numbers by Common Factor

Adding Numbers by Common Factor — Moosa Academy

When every number in a sum shares a factor, pulling it out replaces a row of awkward numbers with a row of small ones. In a fraction the effect is stronger still: the shared factor appears above and below, and simply cancels.

Concept The distributive law, in reverse

Expanding turns $12(1 + 2)$ into $12 + 24$ . Reading that backwards is the whole technique:

12 + 24 = 12 \times 1 + 12 \times 2 = 12(1 + 2)

If every term of a sum is divisible by the same number, write that number outside a bracket and put the quotients inside. The sum inside is always easier than the one you started with.

Example A sum of multiples

Find $12 + 24 + 36 + 48 + 60$ .

Every term is a multiple of

12

Take it out:

12(1 + 2 + 3 + 4 + 5)

The bracket adds to

15

Multiply back:

12 \times 15

$180$

Five two-digit additions have become one small addition and one multiplication. The bracket is often a consecutive sum, which has its own formula if it runs long.

Example Cancelling in a fraction

Compare $\dfrac{18 + 27 + 45}{36 + 9}$ with $2$ .

Every number above and below is a multiple of

9

Factor both:

\dfrac{9(2 + 3 + 5)}{9(4 + 1)}

The

9

cancels:

\dfrac{2 + 3 + 5}{4 + 1} = \dfrac{10}{5}

Divide:

2

The two values are equal

Adding the numerator and denominator directly would give $\dfrac{90}{45}$ — the same answer, but with larger numbers to handle. Cancelling first keeps everything small.

Example When only the top factorises

Compare $\dfrac{6 + 12 + 18 + 24}{12}$ with $7$ .

The numerator shares a factor of

6

Take it out:

\dfrac{6(1 + 2 + 3 + 4)}{12}

The bracket adds to

10

\dfrac{6 \times 10}{12} = \dfrac{60}{12}

Divide:

5

$5 < 7$ , so the second value is larger

Here the factor $6$ does not cancel with the $12$ outright — it halves it. Factoring still pays off, because $6 \times 10$ is much less trouble than adding $6 + 12 + 18 + 24$ in your head.

Note Finding the factor

Look at the smallest number in the sum first — the shared factor can never exceed it, and very often is it.

12 + 24 + 36 + 48 + 60

, the smallest is

12

, and it divides all the rest.

18 + 27 + 45

, the smallest is

18

, which does not work — but

9

does.

When the smallest term itself fails, test its factors in turn. Any common divisor helps; the largest one helps most.

Summary

Taking out a common factor is the distributive law read backwards.
If every term shares a divisor, write it outside a bracket with the quotients inside.
In a fraction, factor the numerator and denominator and cancel the shared factor.
Even when it does not cancel completely, factoring makes the arithmetic lighter.
Start the search at the smallest term; if it fails, try its factors.
The bracket that remains is often a consecutive sum you can evaluate with a formula.