What Is the Greatest Common Factor of 90 and 135? | Proof

The greatest common factor is 45, because 45 is the largest whole number that divides both 90 and 135 with no remainder.

If you’re asking, “What Is the Greatest Common Factor of 90 and 135?”, you’re trying to find the biggest number that fits evenly into both. Once you know that number, you can shrink fractions, simplify ratios, and split things into equal groups without leftovers. This page walks through three reliable ways to get the answer, shows why they all land on the same result, and gives you a couple of quick checks so you can trust the math.

What A Greatest Common Factor Means In Plain Math

A factor is a whole number that divides another whole number evenly. A common factor is a factor shared by two numbers. The greatest common factor (often written as GCF) is the largest shared factor.

So the task is simple: list the shared factors of 90 and 135, then pick the biggest one. The trick is doing it without wasting time.

Start With A Fast Reality Check

Before you do any heavy lifting, it helps to eyeball a couple of shared factors:

• Both numbers end in 0 or 5, so 5 divides both.
• Both numbers are even? No. So 2 can’t be a shared factor.
• The digit sums are 9 and 9, so 9 divides both (since 9 is divisible by 9).

That already tells you the GCF is at least 9, and it must be a multiple of 5 only if both numbers share 5 at the same time. Keep those clues in your pocket. Next, we’ll nail the exact answer.

What Is the Greatest Common Factor of 90 and 135? With Prime Factors

Prime factorization is the “build it from primes” approach. You break each number into prime factors, then keep only the primes they share, using the smallest counts.

Prime Factorization Of 90

Start dividing by small primes:

• 90 = 2 × 45
• 45 = 3 × 15
• 15 = 3 × 5

Put it together: 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5.

Prime Factorization Of 135

Do the same:

• 135 = 5 × 27
• 27 = 3 × 9
• 9 = 3 × 3

So 135 = 3 × 3 × 3 × 5 = 3³ × 5.

Build The GCF From Shared Primes

Now line up the primes:

• 90 has 2 × 3² × 5
• 135 has 3³ × 5

Shared primes are 3 and 5. Use the smaller power of 3, which is 3². That gives:

GCF = 3² × 5 = 9 × 5 = 45.

Why All Three Methods Agree

Once numbers are written as prime factors, every factor is just a product of some of those primes. A factor shared by both numbers must use only primes that appear in both lists. Picking the largest shared factor means taking as many shared primes as you can without exceeding either number’s supply. That’s exactly what “use the smaller exponent” does.

If you like seeing the rule stated cleanly, Khan Academy’s explanation of greatest common divisor and greatest common factor matches this idea with worked examples.

Euclid’s Algorithm For 90 And 135

Euclid’s algorithm sounds fancy, but it’s just a repeatable division pattern. You divide the larger number by the smaller, keep the remainder, then repeat using the smaller number and that remainder. When the remainder becomes 0, the last nonzero remainder is the GCF.

Run The Divisions

Start with 135 and 90:

• 135 ÷ 90 = 1 remainder 45 (since 135 = 1×90 + 45)
• 90 ÷ 45 = 2 remainder 0 (since 90 = 2×45 + 0)

Remainder hit 0, so the GCF is the last remainder before 0, which is 45.

Why This Works Without Any Prime Factors

If a number divides both 135 and 90, it also divides their difference, 135 − 90 = 45. That means the shared factors of (135, 90) are the same shared factors of (90, 45). Repeating that step keeps the shared factor set the same, while the numbers get smaller. That’s the whole reason Euclid’s algorithm ends at the GCF.

OpenStax explains this same divisor idea in its free text while working through factoring and divisibility topics; see the relevant section in Prealgebra: Find The Greatest Common Factor And Least Common Multiple.

Table Of Common GCF Routes And When To Use Each

Route	What You Do	When It Helps
List factors	Write all factors of each number, then match	Small numbers, quick classroom checks
Prime factorization	Break into primes, multiply shared primes with smallest counts	Medium numbers, clean “show your work” steps
Euclid’s algorithm	Repeat division with remainders until remainder hits 0	Large numbers, fastest by hand with practice
Common multiples link	Use LCM × GCF = product (for two numbers)	When you already found the LCM
Shared prime blocks	Group primes into chunks (like 9 and 5 here)	Fast mental math after factoring
Division test ladder	Divide both numbers by shared primes step by step	When you want GCF and reduced numbers together
Factor tree	Draw splits until all leaves are prime	Visual learners, avoiding missed primes
Calculator check	Use gcd() function to confirm your manual work	Homework verification, not as the only step

Factor Listing Method (And A Trick To Keep It Short)

Listing factors can be slow if you try to write everything in order. The shortcut is to list factor pairs. Each pair multiplies to the target number, so you only need to test up to the square root.

Factor Pairs Of 90

• 1 × 90
• 2 × 45
• 3 × 30
• 5 × 18
• 6 × 15
• 9 × 10

So the factor list includes 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

Factor Pairs Of 135

• 1 × 135
• 3 × 45
• 5 × 27
• 9 × 15

So the factor list includes 1, 3, 5, 9, 15, 27, 45, 135.

Pick The Largest Shared One

Compare the lists: the shared factors are 1, 3, 5, 9, 15, 45. The biggest is 45.

Table Of Shared Factors And What They Let You Do

Shared Factor	Shows Up As	What It Lets You Simplify
3	Both divisible by 3	Basic fraction reduction
5	Both end in 0 or 5	Ratios with 5-based groups
9	Digit sums are multiples of 9	Quick checks for shared divisibility
15	3 × 5 shared	Split into 15 equal parts
45	9 × 5 shared	Largest clean split for both numbers

How The Answer Helps With Fractions And Ratios

The GCF is most useful when you want a reduced form. Take the fraction 90/135. Divide top and bottom by 45:

• 90 ÷ 45 = 2
• 135 ÷ 45 = 3

So 90/135 reduces to 2/3. No guessing, no trial reductions, just one clean step.

The same move works for ratios. If a recipe uses a 90:135 ratio, dividing both parts by 45 turns it into 2:3. That’s easier to scale up or down.

A Quick Mental Pattern For Numbers Like These

When both numbers end in 0 or 5, start by pulling out 5. Then check if the remaining parts share 3, 9, or 15. Here’s that pattern on this pair:

• 90 = 5 × 18
• 135 = 5 × 27

Now the job becomes finding the GCF of 18 and 27. Many people spot 9 right away, since 18 = 9×2 and 27 = 9×3. Multiply back by 5, and you get 45.

Mistakes That Trip People Up

Most errors come from one of these slips:

• Stopping at a shared factor that isn’t the largest. Seeing that both are divisible by 9 and quitting there is common. Always ask, “Can I still divide both by something?”
• Mixing up LCM and GCF. The least common multiple is bigger than both numbers (unless they match). The greatest common factor is never bigger than the smaller number.
• Losing a prime in the factorization. On 90, it’s easy to miss that 45 is 3×15 and 15 is 3×5, giving two 3s total.
• Remainder arithmetic errors. In Euclid’s algorithm, one wrong remainder breaks everything. Write the equation line as a check.

Mini Practice Set With Answers

Try these pairs using the same steps. After each, compare with the answer so you can spot where a mistake would show up.

Practice 1: 24 And 60

Prime factors: 24 = 2³×3, 60 = 2²×3×5, so GCF = 2²×3 = 12.

Practice 2: 48 And 180

48 = 2⁴×3, 180 = 2²×3²×5, so GCF = 2²×3 = 12.

Practice 3: 98 And 147

98 = 2×7², 147 = 3×7², so GCF = 7² = 49.

A Short Checklist You Can Reuse

• Check easy shared divisors (2, 3, 5, 9, 10) to set a lower bound.
• Pick a main route: prime factors for clarity, Euclid for speed.
• After you get a candidate, divide both numbers by it to confirm a remainder of 0.
• Try one more step: see if both quotients still share a whole-number factor. If they do, your candidate wasn’t the greatest.

Final Verification For 90 And 135

Test 45 directly:

• 90 ÷ 45 = 2 with remainder 0
• 135 ÷ 45 = 3 with remainder 0

Try any number bigger than 45. It can’t divide 90, since 90 has no factor between 45 and 90 except 90 itself, and 90 doesn’t divide 135 evenly. So 45 is the greatest shared factor.