Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
Saying that 36 is divisible by 4 means the division comes out exact: 36 = 4 × 9, remainder zero. We also say that 4 is a divisor of 36, or that 36 is a multiple of 4. Every number hides a collection of divisors: those of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
To spot divisors without setting out a division, express tests exist. Divisible by 2: the number ends in 0, 2, 4, 6 or 8. By 5: it ends in 0 or 5. By 3: the sum of its digits is divisible by 3 (for 741: 7 + 4 + 1 = 12, bingo!). By 9: the sum of its digits is divisible by 9.
When two numbers share divisors, the largest one is called the GCD (greatest common divisor). To find it, Euclid's algorithm is unbeatable: divide the bigger number by the smaller, then the divisor by the remainder, and repeat until the remainder is zero. The last nonzero remainder is the GCD. It is one of the oldest algorithms in the world, and it still runs inside our computers!
What is the GCD for? Simplifying fractions in one go. Dividing the numerator and denominator of 84/30 by their GCD 6 gives 14/5: a fraction in lowest terms, impossible to simplify any further. A fraction is in lowest terms exactly when the GCD of its numerator and denominator is 1.
Compute the GCD of 84 and 30 using Euclid's algorithm.
84 = 30 × 2 + 24
I divide the bigger number by the smaller: quotient 2, remainder 24.
30 = 24 × 1 + 6
I repeat with 30 and 24: the remainder is 6.
24 = 6 × 4 + 0
I repeat with 24 and 6: zero remainder, I stop. The last nonzero remainder is 6.
Result: GCD(84, 30) = 6
Reduce the fraction 84/30 to lowest terms.
PGCD(84 ; 30) = 6
I reuse the GCD found with Euclid's algorithm: 6.
84 ÷ 6 = 14 et 30 ÷ 6 = 5
I divide the numerator and the denominator by 6, in a single step.
PGCD(14 ; 5) = 1
14 and 5 share no divisor other than 1: the fraction 14/5 is in lowest terms.
Result: 84/30 = 14/5