Indices and number theory

Index notation

In $5^3$, 5 is the base and 3 is the index: $5^3 = 5 \times 5 \times 5$. Exams ask for conversions both ways — writing $2 \times 5 \times 3 \times 5 \times 2$ in index notation means grouping equal factors: $2^2 \times 3 \times 5^2$.

The index laws

With the same base:

• Multiply: add the indices — $a^m \times a^n = a^{m+n}$
• Divide: subtract the indices — $a^m \div a^n = a^{m-n}$
• Power of a power: multiply the indices — $(a^m)^n = a^{mn}$

They combine: $(5^6 \div 5^3)^3 = (5^3)^3 = 5^9$. When bases differ, rewrite one so they match: $3^5 \times 9^4 = 3^5 \times (3^2)^4 = 3^5 \times 3^8 = 3^{13}$.

Prime factorisation

A factor tree breaks a number into primes; keep splitting until every branch ends in a prime, then write the result in index form: $576 = 2^6 \times 3^2$.

Prime factors do heavy lifting:

• Square roots: halve every index — $\sqrt{576} = 2^3 \times 3 = 24$.
• Matching forms: $375 = 3 \times 5^3$ answers “find $p$ and $q$ if $375 = 3^p \times 5^q$”.

HCF and LCM

Write each number as a product of primes first.

• HCF: take the common primes, each to its lowest power.
• LCM: take every prime that appears, each to its highest power.

For $42 = 2 \times 3 \times 7$ and $90 = 2 \times 3^2 \times 5$: HCF $= 2 \times 3 = 6$ and LCM $= 2 \times 3^2 \times 5 \times 7 = 630$.

Divisibility rules

• 2: even. 5: ends in 0 or 5. 10: ends in 0.
• 3: digit sum divisible by 3. 9: digit sum divisible by 9.
• 4: last two digits divisible by 4. 8: last three digits divisible by 8.
• 6: divisible by both 2 and 3.

“Explain why” questions want the rule stated and applied: 630 is divisible by 6 because it is even and $6 + 3 + 0 = 9$ is divisible by 3. The rules also spot composites quickly — a long repunit like 111 is composite because its digit sum is divisible by 3.