Factors and Multiples

Read divisibility in the correct direction

For integers $d\ne0$ and $n$, the notation $d\mid n$ means that $n=dk$ for some integer $k$. Read it as “d divides n”. The vertical stroke is a statement of divisibility, not a fraction bar. For example,

By contrast, $5\nmid36$ because no integer multiplied by 5 produces 36. The fact that $36/5=7.2$ exists does not make 5 an integer factor. Factors in this lesson are integer factors; allowing arbitrary real factors would answer a different question.

When a school problem asks for “the factors of 36”, it usually means its positive factors. We will state that convention in factor lists. Negative factors also exist: $36=(-4)(-9)$.

Find every factor by pairing divisors

To list all positive factors of 36, test positive integers in increasing order and keep a pair whenever division is exact:

The value 5 produces a remainder, so it contributes no pair. Collect the distinct members of the successful pairs:

The pair $(6,6)$ contributes one distinct factor, not two. Listing pairs first reduces the chance of overlooking a larger partner. This matches the systematic factor-list approach in OpenStax’s treatment of factors.

Explain why the search can stop

If $n=ab$ with positive integers $a$ and $b$, at least one factor is no larger than $\sqrt n$. If both were larger, their product would exceed $n$, a contradiction. Testing through $\sqrt n$ therefore finds a member of every factor pair.

For 36, the stopping point is 6. Testing 9, 12 and larger factors would only rediscover pairs already recorded in reverse. For a non-square, stop at the greatest integer below its square root.

This also explains a useful pattern. A positive integer has an odd number of positive factors exactly when it is a perfect square. Factors normally occur in unequal pairs; a square has one central pair with the same member twice. The number 36 has nine positive factors, including its unpaired central factor 6.

Generate multiples without changing the group size

The positive multiples of 6 are $6,12,18,24,\ldots$, produced by multiplying 6 by $1,2,3,4,\ldots$. If all integer multiples are allowed, the list extends in both directions and includes zero:

Zero is a multiple of every non-zero integer because $d\times0=0$. There are infinitely many multiples of a non-zero integer, but only finitely many positive factors of a positive integer.

The phrase “first five multiples” often means the first five positive multiples. State whether zero is being included rather than relying on that phrase alone. The multiples of zero form the set $\{0\}$, not an increasing list of different numbers.

Use factors to organise an exact total

Suppose 36 identical tiles must be placed in equal rows, with at least four and at most nine tiles per row. The permitted row sizes are the factors of 36 in that interval: 4, 6 and 9.

The arrangements are 9 rows of 4, 6 rows of 6, or 4 rows of 9. A row size of 8 would leave four tiles after four full rows, so it fails the equal-row requirement. Rotating an arrangement exchanges rows and columns without changing the total.

This is a divisibility problem before it is a drawing problem. A sketch helps interpret the result, but checking that rows multiplied by tiles per row equals 36 establishes the total.

Connect common factors and common multiples

The positive common factors of 12 and 18 are $1,2,3,6$. They divide both totals. Their positive common multiples begin $36,72,108,\ldots$; each of these totals can be divided into groups of either 12 or 18.

The Greatest Common Factor chooses the largest shared divisor. The Least Common Multiple chooses the smallest positive shared multiple. These are different questions, even though both involve the same two numbers.

Related topics

For a broader introduction to number systems, use What are Numbers?. Arithmetic Operations reviews the multiplication and division used here.

Use Prime Numbers to identify numbers with only two positive factors, then Prime Factorization to describe the structure behind a full factor list.