L6 · Divisibility through prime factors and powers

Learning goal

“I can decide whether one number is a factor of another — without dividing — by comparing their prime factorizations.”

Divisible by — a number is divisible by a divisor when the division leaves no remainder. 48 is divisible by 6 because 48 ÷ 6 = 8 exactly.

Prime factor — a prime number inside a factorization, like the 3 in 48 = 2⁴ × 3.

Factor (of a number) — a whole number that divides it exactly. Asking “is 12 a factor of 48?” is the same question as “is 48 divisible by 12?”

Worked example

Put 48 = 2 × 2 × 2 × 2 × 3 (2⁴ × 3) on trial. Divisible by 6? 6 = 2 × 3 — both primes are inside 48 → yes (48 ÷ 6 = 8). Divisible by 5? 48 has no prime factor 5 → no, without dividing anything. Divisible by 8? 8 = 2 × 2 × 2 needs three 2s; 48 has four → yes (48 ÷ 8 = 6). Say the rule plainly: the divisor's prime factors — every one of them, repeats included — must all fit inside the number's prime factorization.

Check yourself — explain it

48 is NOT divisible by 9 because…

Paired problem — example still visible

Is 48 divisible by 12? → Yes — 12 = 2 × 2 × 3 fits inside 2⁴ × 3 (48 ÷ 12 = 4)

Faded completion — last step is yours

“Is 48 divisible by 18? 18 = 2 × 3 × 3. 48 has the 2, but only one ___, so 48 is ___ divisible by 18.” → 3; not

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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