Number: Operations and Primes

All numbers can be placed on a number line. The six comparison symbols are:

Symbol	Meaning	Example
$=$	equal to	$0.5=\frac{1}{2}$
$\ne$	not equal to	$0.5\ne \frac{1}{3}$
$<$	less than	$-3<-1$
$>$	greater than	$\frac{3}{4}>0.7$
$\le$	less than or equal to	$x\le 5$
$\ge$	greater than or equal to	$x\ge -2$

To compare a mixture of fractions, decimals and integers, convert everything to decimals first.

Worked example — place in ascending order: $\frac{3}{5},0.62,\frac{2}{3},0.6$

Convert to decimals: $\frac{3}{5}=0.60$; $\frac{2}{3}=0.6\overline{6}$

Ascending order: $0.6<\frac{3}{5}<0.62<\frac{2}{3}$ ✓

For negative numbers: further left on the number line means smaller. $-5<-2<0<3$.

Place value determines what each digit in a number is worth. In 3,274.56, the digit 3 is worth 3,000; the digit 5 is worth $\frac{5}{10}=0.5$.

The four operations — add, subtract, multiply, divide — apply to all number types: integers, decimals, fractions (proper, improper, and mixed numbers), both positive and negative.

Adding/subtracting fractions — find a common denominator: $\frac{3}{4}+\frac{2}{3}=\frac{9}{12}+\frac{8}{12}=\frac{17}{12}=1\frac{5}{12}$

Multiplying fractions — multiply numerators, then denominators; simplify: $\frac{3}{4}\times \frac{2}{5}=\frac{6}{20}=\frac{3}{10}$

Dividing fractions — keep the first fraction, change ÷ to ×, flip the second (its reciprocal): $\frac{3}{4}÷\frac{2}{5}=\frac{3}{4}\times \frac{5}{2}=\frac{15}{8}=1\frac{7}{8}$

Mixed numbers — convert to improper fractions before any operation: $2\frac{1}{3}=\frac{7}{3}$.

Formal written methods are structured column techniques that Edexcel expects students to apply reliably.

Column addition/subtraction — align digits by place value; carry or borrow across columns:

$\begin{array}{r}3476\\ +895\\ 4371\end{array}$

For decimals, align the decimal points: $12.47+8.3=12.47+8.30=20.77$ ✓

Long multiplication — split the multiplier by place value:

$384\times 27=384\times 20+384\times 7=7680+2688=10368$

For decimals: $3.84\times 2.7$ — ignore decimal points, compute $384\times 27=10368$, then place the decimal: 2 decimal digits in $3.84$, 1 in $2.7$ gives 3 total → $10.368$ ✓

Long division — worked example: $2952÷8$

$8\text{ into }29=3\text{ remainder }5;8\text{ into }55=6\text{ remainder }7;8\text{ into }72=9$

Result: $369$ ✓. Check: $369\times 8=2952$ ✓

For non-integer quotients, continue past the decimal point (bringing down zeros): $75÷4=18.75$.

Operations must be performed in a specific order. BIDMAS (also written BODMAS) sets this out:

Letter	Operation	Priority
B	Brackets	1st — work inside brackets first
I	Indices (powers and roots)	2nd
D/M	Division and Multiplication	3rd — equal priority, work left to right
A/S	Addition and Subtraction	4th — equal priority, work left to right

Worked example — calculate $3+4\times (5-2{)}^2÷6$:

1. Brackets: $5-2=3$ → $3+4\times 3^2÷6$
2. Indices: $3^2=9$ → $3+4\times 9÷6$
3. Multiplication then division (left to right): $4\times 9=36$, then $36÷6=6$ → $3+6$
4. Addition: $3+6=9$ ✓

Inverse operations undo each other: addition ↔ subtraction; multiplication ↔ division; squaring ↔ square root. Used to rearrange equations and verify answers.

The reciprocal of $x$ is $\frac{1}{x}$. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. Any number multiplied by its reciprocal equals 1.

A prime number has exactly two factors: 1 and itself. The primes start: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …

1 is not prime — it has only one factor. 2 is the only even prime.

Term	Definition	Example
Factor (divisor)	Divides a number exactly with no remainder	Factors of 12: 1, 2, 3, 4, 6, 12
Multiple	Result of multiplying a number by any positive integer	Multiples of 5: 5, 10, 15, 20 …
Common factor	A factor shared by two or more numbers	Common factors of 12 and 18: 1, 2, 3, 6
Common multiple	A multiple shared by two or more numbers	Common multiples of 4 and 6: 12, 24, 36 …
HCF	Highest Common Factor — largest common factor	HCF(12, 18) = 6
LCM	Lowest Common Multiple — smallest positive common multiple	LCM(4, 6) = 12

Every integer greater than 1 can be written as a product of prime factors in exactly one way — the Unique Factorisation Theorem.

Factor tree method — worked example: write 360 as a product of prime factors.

Split 360 repeatedly into factor pairs until all branches end in primes:

$360=2\times 180=2\times 2\times 90=2\times 2\times 2\times 45=2\times 2\times 2\times 3\times 15=2\times 2\times 2\times 3\times 3\times 5$

Written in index notation: $360=2^3\times 3^2\times 5$

Check: $8\times 9\times 5=8\times 45=360$ ✓

The order of splits in the factor tree does not matter — the final product of prime factors is always the same.

Prime factorisation is the most reliable method for finding HCF and LCM of larger numbers.

Worked example — find the HCF and LCM of 72 and 120.

Step 1 — factorise both: $72=2^3\times 3^{2}120=2^3\times 3\times 5$

Step 2 — HCF (highest common factor): take the lowest power of each prime that appears in both: $\text{HCF}=2^3\times 3^1=8\times 3=24$

Step 3 — LCM (lowest common multiple): take the highest power of each prime that appears in either: $\text{LCM}=2^3\times 3^2\times 5=8\times 9\times 5=360$

Verification: for any two positive integers, $a\times b=\text{HCF}(a,b)\times \text{LCM}(a,b)$.

Check: $72\times 120=8640$ and $24\times 360=8640$ ✓

1. Including 1 as a prime number

1 has only one factor (itself), so it does not meet the definition. Prime numbers must have exactly two distinct factors: 1 and the number itself.

2. BIDMAS division/multiplication order

Division and multiplication have equal priority — work left to right. $12÷4\times 3=3\times 3=9$, not $12÷12=1$.

3. HCF vs LCM confusion

HCF uses the lowest powers of shared primes (what fits into both numbers). LCM uses the highest powers of all primes that appear (what both numbers divide into). As a quick check: HCF ≤ smaller number ≤ larger number ≤ LCM.

4. Not converting mixed numbers before operating

$2\frac{1}{3}\times 1\frac{1}{2}$ requires converting to $\frac{7}{3}\times \frac{3}{2}=\frac{21}{6}=\frac{7}{2}=3\frac{1}{2}$ — operating directly on the whole and fractional parts separately gives the wrong answer.

Mistake	Correction
"1 is a prime number"	1 is neither prime nor composite — exactly two factors required
"$12÷4\times 3=1$"	Left-to-right: $12÷4=3$, then $3\times 3=9$
"HCF of 12 and 18 is 36"	36 is the LCM; HCF is 6 (highest factor in both)