Reduce a fraction to lowest terms, with both routes to the divisor shown and compared.
Reduce to lowest terms
Both routes to the divisor.
48/180
4/15
Divided top and bottom by 12.
Greatest common divisor
12
shared primes: 2² × 3
Mixed number
4/15
proper — no whole part
Decimal
0.2(6)
repeats, block of 1
Euclid took
4 lines
and never factorised anything
Route 1 — Euclid’s algorithm
48 = 0 × 180 + 48
180 = 3 × 48 + 36
48 = 1 × 36 + 12
36 = 3 × 12 + 0
The last non-zero remainder is the divisor. Nothing here needed a factorisation, which is why this route still works on six-digit numbers.
Route 2 — prime factorisation
Prime factorisation of numerator and denominator with the shared part
Numerator
2⁴ × 3
Denominator
2² × 3² × 5
In common
2² × 3
Each shared prime taken to the LOWER of its two powers. That product is the same 12Euclid found — two methods, one number.
Or one prime at a time
48/180 ÷2→ 24/90 ÷2→ 12/45 ÷3→ 4/15
Same destination, 3 steps instead of one. Cancelling by the greatest common divisor is simply doing all of them at once.
The same number, written larger
4/15 = 8/30 = 12/45 = 16/60 = 20/75 = …
Every one of these is the same value. Lowest terms is the one place the chain stops, which is why it is the canonical form.
Euclid took 4 lines and never factorised anything; the prime route needed both factorisations first. On numbers this size either is fine — on six-digit ones only Euclid is.
Both routes are computed independently and compared — if they ever disagreed, this line would say so.
What this tool shows
The greatest common divisor is found twice, by unrelated methods, and the two must agree before anything is shown. A fraction that is already in lowest terms is reported as such rather than treated as a failure.
The greatest common divisor by Euclid’s algorithm
The same divisor read off the shared prime factors
Cancelling one prime at a time, as an alternative
A negative denominator moved to the numerator
The answer as a mixed number and an exact decimal
A proof, not a shrug, when nothing can be cancelled
Euclid, line by line Prime factorisation beside it Both computed, then compared Mixed number and exact decimal
Updated 7 September 2026 · Works in any browser, no installation
To simplify a fraction, divide the numerator and the denominator by their greatest common divisor. That single division finishes the job — cancelling by 2 twice and then by 3 arrives at the same place more slowly. 48/180 has a greatest common divisor of 12, so it reduces in one step to 4/15, and 4 and 15 share no factor above 1, which is what makes it lowest.
At a glance
Formula shown
a/b in lowest terms is (a \u00f7 g)/(b \u00f7 g) where g = gcd(a, b). Euclid: gcd(a, b) = gcd(b, a mod b), until the remainder is zero. By primes: g is the product of every shared prime raised to the lower of its two powers.
Scenario support
Marking or checking homework where the reduction step is the answer; putting a measurement into its simplest form; reducing an odds or ratio before comparing it with another.
Educational estimate
Planning support from the values you enter — not professional advice.
Why two routes to the same number
Euclid’s algorithm and prime factorisation both produce the greatest common divisor, and they are good at different things. Euclid is what a computer does: it never factorises anything, it just takes remainders, and it finishes in a handful of lines even on six-digit numbers where factorising would be hopeless.
Prime factorisation is what a classroom asks for, and it is the one that shows where the answer comes from. Seeing that 48 is 2⁴ × 3 and 180 is 2² × 3² × 5 makes it obvious that the shared part is 2² × 3 — each prime taken to the LOWER of its two powers — and that this is 12.
This page runs both and compares them. If they ever disagreed the page would say so; that they always agree is a fact about the mathematics, not an assumption in the code.
Euclid, in three lines
The algorithm is one idea repeated: the greatest common divisor of two numbers is the same as the divisor of the smaller one and the remainder. Keep replacing the pair until the remainder is zero, and the last non-zero remainder is the answer.
For 1071 and 462 that is three lines: 1071 = 2 × 462 + 147, then 462 = 3 × 147 + 21, then 147 = 7 × 21 + 0. The divisor is 21. Nothing was factorised on the way, which is why the same three lines would work if the numbers had twenty digits.
It is roughly two-and-a-half thousand years old and still the method every computer algebra system uses, which is a reasonable argument for learning it.
The shared primes
Every whole number above one factorises into primes in exactly one way. Once both numbers are written that way, the common part is read straight off: take every prime that appears in both, raised to whichever power is smaller.
The word LOWER matters. 2⁴ in the numerator and 2² in the denominator gives 2² in the divisor, not 2⁴ — you can only cancel as many twos as the shorter side has.
The tool also shows the cancellation one prime at a time, which is what you would do by hand without calculating the divisor first. Same destination, more steps — and it makes clear that dividing by the greatest common divisor is simply doing all of them at once.
When it is already reduced
7/13 cannot be simplified. That is not the tool failing to find something; it is a positive result, and it is worth stating clearly: 7 and 13 are both prime and different, so they share nothing but 1.
This matters when you are checking work. “Simplify 7/13” has the answer “7/13”, and a student who writes that has done the question correctly. A tool that returns an error, or silently echoes the input without comment, leaves it ambiguous.
The page says which it is every time: either what it divided by, or that there was nothing to divide by and why.
The sign belongs on top
12/−18 and −12/18 and −(12/18) are the same number. Only one of them is written the way every later rule assumes, and that is the one with the sign on the numerator.
It is not fussiness. Comparing fractions by cross multiplication — a/b against c/d by comparing ad with cb — is only valid while b and d are both positive. Leave a negative denominator in place and the comparison silently reverses, with nothing to warn you.
So this page moves the sign first and says that it did. Everything afterwards is then safe.
Lowest terms is a canonical form
4/15, 8/30, 12/45, 16/60 and infinitely many more are all the same number. Only one of them is in lowest terms, and that is what makes lowest terms useful: it is the one form two people working separately will both arrive at.
That is why answers are expected in it, why fractions are stored that way internally on this site, and why two fractions can be compared for equality just by comparing their reduced pairs — no arithmetic needed.
The tool shows a few of the larger equivalents beneath the answer, to make the point that the chain runs upwards forever and stops in exactly one place going down.
Sources and methodology
Reduction is settled mathematics. These are where the algorithm and the convention are set out.
Method. The divisor is computed twice. Euclid’s algorithm runs on the pair and every line of it is recorded, including the quotient and remainder, so the working can be read rather than trusted. Separately both numbers are factorised by trial division and the shared primes are taken to their lower powers. The two results are compared before anything is displayed, and the suite below asserts they agree for every fraction up to sixty over sixty. That engine is verified on every change against 58 hand-written assertions, including that Euclid and the prime route return the identical divisor across seven thousand fractions, and that the reduced pair always shares no factor above one. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
FractionAdd, subtract, multiply and divide fractions and mixed numbers exactly, with the least common denominator chosen, the cross-cancelling done first, and the true repeating decimal beside the answer.
GCFThe greatest common factor of two to six numbers with all three routes shown — the shared primes to their lower powers, Euclid line by line, and the full factor lists when they are short enough to be honest.
Fraction to DecimalConvert a fraction to its exact decimal, with the length of the repeat worked out from the denominator before dividing and the long division shown remainder by remainder.
Comparing FractionsOrder up to eight fractions with all three methods shown on your own numbers, and a figure for how many decimal places the closest pair need before they stop looking equal.
Mixed NumberAdd, subtract, multiply and divide mixed numbers with the borrow shown where it happens, plus a second independent check worked through improper fractions.
Decimal to FractionTurn any decimal into its exact fraction, repeating ones included, with the four algebra lines that cancel the infinite tail instead of rounding it away.
If you need to combine fractions rather than reduce one, the Fraction Calculator does the four operations and reduces at the end anyway. If you want the greatest common factor of two plain numbers rather than a fraction, the GCF Calculator is the same algorithm without the fraction around it.
Educational use disclaimer
This is an educational tool. The reduction is exact and independently verified by two separate methods, but the presentation follows one common convention — your course may write the working differently for the same answer.
Published the simplify fractions page: the greatest common divisor computed twice, by Euclid and by prime factorisation, and compared before anything is shown.
A fraction already in lowest terms is reported as such with the reason, rather than being echoed back or treated as an error.
A negative denominator is moved onto the numerator first and the page says so, because cross multiplication silently reverses without that step.
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