Adding Numbers by Common Factor

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

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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 into . Reading that backwards is the whole technique:

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 .

Every term is a multiple of
Take it out:
The bracket adds to
Multiply back:

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 with .

Every number above and below is a multiple of
Factor both:
The cancels:
Divide:
The two values are equal

Adding the numerator and denominator directly would give — the same answer, but with larger numbers to handle. Cancelling first keeps everything small.

Example When only the top factorises

Compare with .

The numerator shares a factor of
Take it out:
The bracket adds to :
Divide:
, so the second value is larger

Here the factor does not cancel with the outright — it halves it. Factoring still pays off, because is much less trouble than adding 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.

In , the smallest is , and it divides all the rest.
In , the smallest is , which does not work — but does.

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

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