A factor is any integer that divides into another integer without leaving a remainder. In mathematical terms, finding the factors of 50 means identifying all integers $n$ where $50 \div n$ results in a whole number quotient.
You do not need to test every integer between 1 and 50. You only need to test numbers up to the square root of 50. Because the square root of 50 is roughly 7.07, testing stops at the whole number 7. Any factor larger than 7 automatically pairs with a factor smaller than 7.
Testing through integer division reveals:
- 50 ÷ 1 = 50 (Remainder 0) → 1 and 50 are factors
- 50 ÷ 2 = 25 (Remainder 0) → 2 and 25 are factors
- 50 ÷ 3 = 16.67 (Remainder 2) → Not a factor
- 50 ÷ 4 = 12.5 (Remainder 2) → Not a factor
- 50 ÷ 5 = 10 (Remainder 0) → 5 and 10 are factors
- 50 ÷ 6 = 8.33 (Remainder 2) → Not a factor
- 50 ÷ 7 = 7.14 (Remainder 1) → Not a factor
Because all values beyond 7 must pair with integers below 7, the positive list ends here. The resulting divisors are 1, 2, 5, 10, 25, and 50.
