Check both the product and completion
A prime is an integer greater than one whose only positive divisors are one and itself. In a prime factorization, every factor is prime. Starting with 84 = 4 × 21, split 4 into 2 × 2 and 21 into 3 × 7. Thus 84 = 2² × 3 × 7. The exponent in 2² means multiplying two copies of 2: 2 × 2.
The product 3 × 4 × 7 also equals eighty-four. However, 4 can still be split into 2 × 2, so that expression is not a completed prime factorization. Compare choices by asking two separate questions: does the product equal the original integer, and is every factor prime? The correct value alone is insufficient.
A common divisor must fit inside both numbers
Write 18 = 2 × 3² and 30 = 2 × 3 × 5 in two rows aligned by prime. Each has one copy of 2. Eighteen has two copies of 3, but thirty has only one. A divisor of both cannot use two copies of 3, because thirty does not supply them. Using the shared counts gives 2 × 3 = 6. The 5 that occurs only in thirty cannot be included.
Check greatest as well. The positive divisors of eighteen are 1, 2, 3, 6, 9 and 18; those of thirty are 1, 2, 3, 5, 6, 10, 15 and 30. Their shared divisors are 1, 2, 3 and 6, so six is greatest. Three divides both but fails the greatest condition. Finally verify 18 ÷ 6 = 3 and 30 ÷ 6 = 5.
A common multiple must contain enough for both
For 12 = 2² × 3 and 18 = 2 × 3², the required number must be divisible by both. Two copies of 2 and two copies of 3 provide enough for each integer. Their product is 2² × 3² = 36. Here each prime needs the larger of the two counts, unlike the smaller shared count used for a common divisor.
Since 36 ÷ 12 = 3 and 36 ÷ 18 = 2, thirty-six is a common multiple. The only positive multiples of twelve below it are twelve and twenty-four, and neither is divisible by eighteen. Therefore thirty-six is least. Seventy-two and two hundred and sixteen are also common multiples, but neither is least. Finding any shared multiple does not finish the task.
Use the two-row note, then return to fractions
On paper, make a column for each prime and write the two example integers in aligned rows of prime counts. For a common divisor, circle the counts that both can supply. For a common multiple, circle the counts needed to contain both. Multiply them, then write two division checks with the original integers. Say which number is the divisor in each check.
Next, follow the related guide “Fractions and ratios: name the whole”. Carry the idea of a common divisor into its explanation of changing numerator and denominator together while preserving the value. If the meaning of division itself is unclear, return to the prerequisite “Everyday calculations: start with the quantity”.