TMUA Number, Factors and Divisibility

Prime factorisation

Every integer above 1 is a unique product of primes, such as $360=2^3\cdot3^2\cdot5$. HCF takes the lowest power of each prime, and LCM takes the highest.

Counting factors

If $n=p^a q^b r^c$, then $n$ has $(a+1)(b+1)(c+1)$ positive factors.

Squares and cubes

A number is a perfect square when every prime power is even, and a perfect cube when every one is a multiple of 3.

Divisibility and consecutive integers

List cases in order when counting, so that no case is missed or repeated.

• A number is divisible by 3 (or 9) if its digit sum is, by 4 if its last two digits are, and by 8 if its last three digits are.
• A number is divisible by 11 if the alternating sum of its digits is.
• Among any three consecutive integers exactly one is a multiple of 3, and among any two one is even.

Worked examples

Common mistakes

• Taking the HCF as the highest power and the LCM as the lowest.
• Forgetting that 1 and the number itself count as factors.
• Treating 1 as a prime.
• Dropping a prime factor during factorisation, so the product no longer matches.

Test yourself on this topic with our free practice questions.