Factors, Multiples and Number Theory
Once you can divide, one question follows immediately: which divisions come out exactly? Everything on this page is an answer to that question, from a seven-year-old sorting counters into equal rows to a proof that the primes never run out.
It is also the most practical branch of arithmetic that does not look practical. Simplifying a fraction is a factor question. Two buses that leave together and meet again is a multiple question. Whether a decimal stops or repeats forever is settled entirely by the prime factors of the denominator.
What is a factor, and what is a multiple?
A factor divides a number exactly; a multiple is what you get by multiplying it. The factors of 12 are 1, 2, 3, 4, 6, 12 and there the list ends. The multiples of 12 are 12, 24, 36, 48, … and that list never ends.
Dividing Exactly is where "exactly" gets pinned down — no remainder, nothing left over — and Factors and Multiples name the two directions. Factor Pairs is the one that makes finding them reliable: factors come in pairs that multiply to the number, so hunting for 12 means checking 1 × 12, 2 × 6, 3 × 4 and then stopping, because after that the pairs repeat in the other order.
Divisibility without dividing
Divisibility Tests collects the checks that answer "does this go?" faster than the division would.
| Divisor | Test | Example |
|---|---|---|
| 2 | last digit is even | 4,718 ✓ |
| 3 | digits add to a multiple of 3 | 4, ✓ |
| 4 | last two digits divide by 4 | 4, ✓ |
| 5 | ends in 0 or 5 | 4,715 ✓ |
| 9 | digits add to a multiple of 9 | 4, ✓ |
The digit-sum tests are the interesting ones, and they are not coincidence: every power of ten leaves a remainder of 1 when divided by 9, so a number and its digit sum leave the same remainder. That fact is worth more than the test — Casting Out Nines to Check Arithmetic turns it into a way of checking a long multiplication in a couple of seconds, and Modular Arithmetic on a Clock Face is the general idea behind it: arithmetic where only the remainder is kept, which is how clocks, calendars and checksums all work.
What is a prime number?
A prime is a whole number above 1 whose only factors are 1 and itself. They are the numbers that cannot be broken up, which makes them the raw material everything else is built from.
The Sieve of Eratosthenes is the oldest way to find them, and it is a genuinely good algorithm rather than a classroom curiosity: write the numbers out, keep 2 and strike out every later multiple of 2, keep 3 and strike out its multiples, and continue. What survives is prime, and you never test a single number for divisibility.
Prime Factorization is the payoff. Every whole number above 1 breaks into primes in exactly one way — — and that unique fingerprint is what makes the next section work.
HCF or LCM — which do you need?
Ask whether you are breaking something down or building it up. Breaking down into equal pieces needs the highest common factor; waiting for two cycles to line up needs the lowest common multiple.
Common Factors and The Highest Common Factor take the first direction: the largest number that divides both. 12 and 18 share 1, 2, 3 and 6, so the HCF is 6 — which is the largest tray you can pack both piles into without a gap.
Common Multiples and The Lowest Common Multiple take the other: the smallest number both divide into. 12 and 18 first meet at 36 — the first time a 12-minute bus and an 18-minute bus leave together again.
Choosing Between HCF and LCM exists because the arithmetic is rarely the hard part; picking the right one from a word problem is. The prime factorizations answer both at once: take the lowest power of each shared prime for the HCF, the highest power of every prime for the LCM.
HCF and LCM from the prime factorizations
For numbers too big to list factors for, the factorizations do the work. Write both as products of primes, then read off both answers from the same pair of rows.
and .
- HCF: the lowest power of each prime they share — 2 × 3 = 6. Only primes appearing in both count, which is why 5 and 7 are ignored.
- LCM: the highest power of every prime that appears in either — ,260.
A useful check falls out of this: multiply the HCF by the LCM and you get the product of the two original numbers, every time. 6 × 1,260 = 7,560 = 84 × 90. Each prime is being counted once at its lower power and once at its higher, which between them is exactly the two numbers' worth.
Do the primes ever run out?
No, and the proof is short enough to hold in your head. Euclid’s Proof That Primes Never Run Out works like this: suppose you had a complete list of every prime. Multiply them all together and add one. The new number leaves a remainder of 1 when divided by every prime on the list — yet every number above 1 has some prime factor. So there is a prime that is not on the list, and the list was never complete.
Which fractions have decimals that stop?
The ones whose denominator, in lowest terms, is built only from 2s and 5s. That is the whole rule, and Why Every Fraction Terminates or Recurs gives the reason: our decimals are built on ten, ten is 2 × 5, and only a denominator that divides some power of ten can be rewritten as tenths, hundredths or thousandths. stops; cannot.
The repeat is also guaranteed rather than merely likely: dividing by 7 can only ever leave remainders 1 to 6, so within seven steps a remainder must repeat, and once a remainder repeats the digits do too.
Turning Recurring Decimals into Fractions runs the process backwards with one trick — multiply by a power of ten to line the repeating block up with itself, then subtract. x = 0.272727…, so 100x = 27.2727…, and 99x = 27, giving x = 27/99 = 3/11.
The same subtraction settles the argument people have about Why 0.999… Equals Exactly One: x = 0.999…, 10x = 9.999…, so 9x = 9 and x = 1. Not almost one. There is no number in between, and two numbers with nothing between them are one number.
What is an irrational number?
A number that cannot be written as a fraction of whole numbers at all. Rational and Irrational Numbers draws the line: every fraction has a decimal that either stops or repeats, so any decimal that does neither is outside the fractions entirely.
The Square Root of Two Is Irrational proves that at least one such number exists, and does it by contradiction again. Write √2 = a/b in lowest terms. Squaring gives a² = 2b², so a² is even, so a is even. Write a = 2c and the same equation becomes b² = 2c², so b is even too. But a fraction in lowest terms cannot have both parts even. The assumption is impossible, so no such fraction exists.
Two older ideas round the branch out. Egyptian Fractions and the Greedy Algorithm looks at a notation that only allowed unit fractions — — and at the greedy method that always terminates, which is a real theorem hiding in a curiosity. Continued Fractions writes a number as a nest of reciprocals and produces the best possible fractional approximations: it is where comes from, a value for π correct to seven digits with three-digit numbers.
The three mistakes worth naming
- Calling 1 prime. It is excluded on purpose: if 1 were prime, prime factorization would stop being unique, since you could paste on any number of 1s.
- Confusing the two lists. Factors are finite and sit below the number; multiples are infinite and sit above it.
- Reading 0.999… as "just under 1". A recurring decimal is not a number creeping towards a limit; it is the limit, written down.
Where this leads next
Factors are the engine under fraction arithmetic — see fractions, equivalence and mixed numbers — and the terminating-or-recurring question hands straight over to the decimals guide. Powers, roots and surds continue the irrational thread in algebraic expressions, indices and surds.
Practise it in the game
Math Challenge teaches each step above as an illustrated lesson — the sieve as a grid you watch fill in, the HCF as a shared factor list, Euclid's proof laid out line by line — inside a catalog of 800+ lessons. For drill, the HCF and LCM topic and the Primes topic run the recognition until it is quick.
Your turn
Three to try — tap what you get.
The highest common factor of 12 and 18
Which of these is prime?
The lowest common multiple of 4 and 6