A factor of a number is any whole number that divides into it evenly, leaving no remainder. Finding "the factors of 60" means finding every whole number that goes into 60 cleanly — and the reliable, by-hand method for doing that is trial division, tested systematically rather than guessed at randomly.
**The method.** Start at 1 and test each whole number in turn: does it divide evenly into your target number? If yes, it's a factor, and so is the number you get by dividing your target by it (its "pair"). Keep going until the number you're testing meets or passes the square root of your target — once you've passed that point, every remaining factor has already turned up as someone else's pair.
Here's why that square-root cutoff works, because it's worth understanding rather than memorizing: factors always come in pairs that multiply to the target number. For 60, the pair (1, 60) multiplies to 60; so does (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). Notice the first number in each pair is climbing (1, 2, 3, 4, 5, 6) while the second is shrinking (60, 30, 20, 15, 12, 10) — they're converging toward the middle. The point where they'd meet is the square root of 60, which is about 7.75. Once your "first number" candidate passes 7.75, you'd just be re-finding pairs you already have, in reverse order. So you never need to test past 8 (rounding up) for 60.
**Worked example: factors of 60.** Test 1: 60 ÷ 1 = 60, no remainder. Factor pair: (1, 60). Test 2: 60 ÷ 2 = 30. Pair: (2, 30). Test 3: 60 ÷ 3 = 20. Pair: (3, 20). Test 4: 60 ÷ 4 = 15. Pair: (4, 15). Test 5: 60 ÷ 5 = 12. Pair: (5, 12). Test 6: 60 ÷ 6 = 10. Pair: (6, 10). Test 7: 60 ÷ 7 = 8.57, not a whole number — 7 is not a factor. Test 8: √60 ≈ 7.75, so 8 is past the cutoff; stop here. Collecting every number from the pairs found: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 — twelve factors total, in order.
**Worked example: factors of 37.** Test 1 through 6 (since √37 ≈ 6.08): only 1 divides evenly (37 ÷ 1 = 37). No other number from 2 to 6 divides into 37 without a remainder. That means 37's only factors are 1 and 37 itself — which is exactly the definition of a prime number. Finding a number has only two factors is, in fact, the same process as checking primality; see the companion guide on checking whether a number is prime.
**A shortcut worth knowing:** you don't need to test every single number one at a time if you recognize a few quick divisibility signals first. A number is divisible by 2 if it's even. It's divisible by 5 if it ends in 0 or 5. It's divisible by 3 if the sum of its digits is divisible by 3 (for 60: 6+0=6, divisible by 3). These quick checks (covered in full in the Divisibility Rules guide) can save you from doing the actual division for numbers that obviously won't divide evenly.
**Common mistakes.** The most frequent error is stopping too early or too late relative to the square root — testing all the way up to the target number itself wastes enormous effort on large numbers (for a number like 200, you'd be testing nearly 200 values instead of about 14), while stopping before the square root risks missing a middle factor pair entirely. A second common mistake is forgetting to record both numbers in a pair — finding that 4 divides into 60 and writing down just "4," forgetting that 15 (60 ÷ 4) is also a factor. A third, more subtle mistake is including a number's negative counterparts by accident when the context calls for positive factors only; unless a problem explicitly asks for negative factors (which do technically exist — −4 also divides evenly into 60), factor lists conventionally include only positive whole numbers.
**Special cases worth knowing.** The number 1 has exactly one factor: itself. Every number has at least two factors if it's prime (1 and itself) or more if it's composite. Perfect squares (like 36, 49, or 100) always end up with an odd total number of factors, because their square root pairs with itself rather than with a different partner — for 36, the pair (6, 6) only contributes the single value 6 to the list, not two separate values, which is why 36 has nine factors (1, 2, 3, 4, 6, 9, 12, 18, 36) instead of an even count.
**Worked example: factors of 100.** √100 = 10 exactly, a clean cutoff. Test 1: 100÷1=100, pair (1,100). Test 2: 100÷2=50, pair (2,50). Test 3: 100÷3=33.3, not whole. Test 4: 100÷4=25, pair (4,25). Test 5: 100÷5=20, pair (5,20). Test 6: not whole. Test 7: not whole. Test 8: not whole. Test 9: not whole. Test 10: 100÷10=10 — here the "pair" is 10 with itself, so it only contributes one value, not two, to the final list. Collecting: 1, 2, 4, 5, 10, 20, 25, 50, 100 — nine factors, an odd count, exactly because 100 is a perfect square and its square root pairs with itself rather than a distinct partner.
**A larger worked example to build confidence with bigger numbers: factors of 144.** √144 = 12 exactly. Test each number 1 through 12: 1 (pair 1,144), 2 (2,72), 3 (3,48), 4 (4,36), 6 (6,24), 8 (8,18), 9 (9,16), 12 (12,12, self-paired). Note 5, 7, 10, and 11 don't divide evenly into 144 and are skipped. Collecting: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 — fifteen factors, again an odd count since 144 is also a perfect square.
Once you're comfortable finding a number's full factor list, the natural next step is prime factorization — breaking a number all the way down into its prime building blocks, which is a related but distinct skill covered in How to Find the Prime Factorization of a Number. And if what you actually need is the largest shared factor between two numbers rather than one number's complete list, see What Is GCF, and How Do You Find It?