OCR GCSE Mathematics: Integer arithmetic, number theory, priority of operations and inverse operations (J560)

This guide is aligned to the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), and covers specification sections 1.01 to 1.04 of Number Operations and Integers: calculations with integers, whole number theory, priority of operations and inverse operations. Everything here is Foundation and Higher content, so it applies to both tiers. A few statements in 1.02b and 1.02c sit in the Foundation column of the specification, and they are marked below. Nothing in this unit is labelled Higher tier only.

For condensed notes and a self-test, use the revision notes. For exam-style questions with full answers, use the practice set. The wider course is at the OCR GCSE Mathematics hub, with a printable checklist. If you are not sure where your gaps are, take a free 10-minute diagnostic first.

What this unit covers

Section	What you must be able to do
1.01a	Add, subtract, multiply and divide positive and negative whole numbers without a calculator
1.02a	Use the terms odd, even, prime, factor (divisor), multiple, common factor, common multiple, square, cube, root; use place value
1.02b	Identify primes under 20; write a whole number as a product of prime factors; know the factorisation is unique. Foundation tier also: identify any prime and use power notation
1.02c	Find the HCF and LCM of two whole numbers by listing. Foundation tier also: find them from prime factorisations
1.03a	Use the conventional order for brackets, the four rules, powers, roots and reciprocals
1.04a	Use addition/subtraction, multiplication/division and powers/roots as inverses, to simplify and check calculations and to reverse “I’m thinking of a number” and “missing digit” problems

Where the specification says “non-calculator methods”, practise by hand. The specification says any content can turn up on a calculator paper or a non-calculator paper, so build both the written methods and the habit of checking.

1.01a Four rules with whole numbers

Adding and subtracting negatives. Think of a number line. Subtracting a negative moves you right, the same as adding.

-8 - (-15) + 6
= -8 + 15 + 6
= 7 + 6
= 13

Multiplying and dividing. Same signs give a positive answer. Different signs give a negative answer. With several factors, count the negatives: an even number gives a positive product.

(-6) x (-7) = 42
56 ÷ (-8)   = -7
(-3) x 4 x (-5) = (-12) x (-5) = 60

Long multiplication by hand. Split the second number into tens and units.

347 x 26
347 x 20 = 6 940
347 x 6  = 2 082
Add:       9 022

Division by hand. Use the bus stop method, or build up multiples of the divisor.

1 548 ÷ 12
12 x 100 = 1 200   leaves 348
12 x 20  =   240   leaves 108
12 x 9   =   108   leaves 0
Answer: 100 + 20 + 9 = 129

The quotient is the result of a division. If the division does not come out exactly, give a remainder: 245 ÷ 12 = 20 remainder 5, because 12 x 20 = 240 and 245 - 240 = 5.

1.02a Terms and place value

Term	Meaning	Example
Odd / even	Not divisible / divisible by 2	17 is odd, 18 is even
Factor (divisor)	Divides exactly into the number	Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Multiple	In the times table of the number	Multiples of 7: 7, 14, 21, 28
Common factor	A factor of two or more numbers	4 is a common factor of 16 and 24
Common multiple	A multiple of two or more numbers	48 is a common multiple of 16 and 24
Prime	Has just two factors: 1 and the number itself	2, 3, 5, 7, 11
Square	A number multiplied by itself	36 = 6²
Cube	A number multiplied by itself twice	27 = 3³
Root	Reverses a power	√36 = 6, ∛27 = 3

Place value. In 40 372, the digit 3 is worth 300, the digit 7 is worth 70, and the digit 4 is worth 40 000. Moving one place left makes a digit worth ten times as much. Place value is what makes the written methods above work: when you multiply by 20 you are multiplying by 2 and then by 10.

1.02b Prime numbers and prime factorisation

A prime has exactly two factors. The number 1 has only one factor, so it is not prime. The number 2 is the only even prime.

The primes under 20 are 2, 3, 5, 7, 11, 13, 17 and 19.

Foundation tier also: identifying any prime. To test a larger number, check for divisibility by each prime up to its square root. If none divides it, the number is prime. Why stop at the square root? Factors come in pairs, and one of each pair is at most the square root.

Is 263 prime?  16 x 16 = 256 and 17 x 17 = 289, so test primes up to 13.
Divisible by 2? No (odd).   By 3? 2+6+3 = 11, no.   By 5? No.
By 7?  7 x 37 = 259, remainder 4. No.
By 11? 11 x 23 = 253, remainder 10. No.
By 13? 13 x 20 = 260, remainder 3. No.
263 is prime.

A number can look prime and not be: 143 = 11 x 13.

Prime factorisation. Divide repeatedly by the smallest prime that fits, or use a factor tree, until only primes remain. From the specification, 24 = 2 x 2 x 2 x 3 and, using power notation, 600 = 2³ x 3 x 5². Here is another.

504 ÷ 2 = 252
252 ÷ 2 = 126
126 ÷ 2 = 63
 63 ÷ 3 = 21
 21 ÷ 3 = 7
  7 is prime

504 = 2 x 2 x 2 x 3 x 3 x 7 = 2³ x 3² x 7

Power notation (a Foundation-column statement, so Foundation tier also) counts repeats: three 2s is 2³, two 3s is 3². Check by multiplying back: 8 x 9 x 7 = 504.

Uniqueness. Any whole number above 1 has one prime factorisation and no other, whatever order you write the factors in. Start the 504 tree with 8 x 63 or 12 x 42 and you still end at 2³ x 3² x 7. That is why prime factorisation is a reliable tool for HCF and LCM.

1.02c HCF and LCM

The highest common factor (HCF) is the largest number that divides into both numbers. The lowest common multiple (LCM) is the smallest number that both numbers divide into.

By listing. List the factors (for HCF) or the first few multiples (for LCM), then pick out the common ones.

16: 1, 2, 4, 8, 16
24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 4, 8     HCF = 8

Multiples of 16: 16, 32, 48, 64
Multiples of 24: 24, 48, 72
LCM = 48

Foundation tier also: from prime factorisations. Write each number as a product of primes. For the HCF, take each shared prime to its lowest power. For the LCM, take every prime that appears, to its highest power.

396 = 2² x 3² x 11
330 = 2 x 3 x 5 x 11

HCF: 2 x 3 x 11            = 66
LCM: 2² x 3² x 5 x 11      = 1 980

A check that works for two numbers: HCF x LCM = product of the numbers. Here 66 x 1 980 = 130 680 = 396 x 330.

Choosing HCF or LCM. If two events repeat and you want to know when they coincide, use the LCM: lights flashing every 14 seconds and every 21 seconds coincide every 42 seconds. If you must split quantities into equal groups with nothing left over, use the HCF: ribbons of 180 cm and 252 cm cut into the longest equal pieces use pieces of 36 cm (HCF = 36).

1.03a Priority of operations

Work in this order:

1. Brackets (and anything that acts like a bracket: a fraction bar, or the line over a root sign).
2. Powers and roots (and reciprocals).
3. Multiplication and division, left to right.
4. Addition and subtraction, left to right.

Multiplication and division have equal priority, so you move left to right. The same goes for addition and subtraction.

4 + 6 x (9 - 5)² ÷ 8
= 4 + 6 x 4² ÷ 8        brackets
= 4 + 6 x 16 ÷ 8        power
= 4 + 96 ÷ 8            multiply (left to right)
= 4 + 12                divide
= 16

48 ÷ 4 x 3 = 12 x 3 = 36      (not 48 ÷ 12 = 4)

Roots and fraction bars. A root sign over a sum works like a bracket.

√(64 - 28) + 3³ = √36 + 27 = 6 + 27 = 33

(17 + 3 x 5) / (2² + 4) = (17 + 15) / (4 + 4) = 32 / 8 = 4

Reciprocals. To find a reciprocal, turn 1 over the number: the reciprocal of 4 is 1/4, and the reciprocal of 1/4 is 4. Work reciprocals out at the powers-and-roots stage.

3 x (reciprocal of 4) x 8 + 5
= 3 x 1/4 x 8 + 5
= 6 + 5
= 11

1.04a Inverse operations

Addition and subtraction undo each other. So do multiplication and division. Squares are undone by square roots, and cubes by cube roots. There are two main uses.

Simplifying a calculation. Change the numbers into friendlier ones, then adjust.

604 - 297 = 604 + 3 - 300 = 307        (adding 3 to 297 gives 300, so add 3 back)
35 x 16 = 70 x 8 = 140 x 4 = 560       (double one factor, halve the other)
45 x 18 = 90 x 9 = 810
1 260 ÷ 15 = 420 ÷ 5 = 84              (divide both numbers by 3)
223 - 98 = 223 + 2 - 100 = 125         (an example from the specification)
25 x 12 = 50 x 6 = 100 x 3 = 300       (another from the specification)

Checking. Reverse the calculation. If 1 548 ÷ 12 = 129, then 129 x 12 must be 1 548. It is: 1 290 + 258 = 1 548.

“I’m thinking of a number.” Write the steps in order, then undo them in reverse order with inverse operations.

I double a number, add 9, then divide by 5. The answer is 7.

Forward:  x2, +9, ÷5  ->  7
Reverse:  7 x 5 = 35,  35 - 9 = 26,  26 ÷ 2 = 13
The number is 13.   Check: 13 x 2 = 26, 26 + 9 = 35, 35 ÷ 5 = 7.

Powers and roots work the same way. “I square a number, subtract 19 and get 45; the number is positive.” Reverse: 45 + 19 = 64, √64 = 8.

Missing digits. Use the inverse operation, or work column by column.

Find the digits a and b in 4a8 + 27b = 705.
Units: 8 + b ends in 5, so b = 7 (carry 1).
Tens: a + 7 + 1 ends in 0, so a = 2 (carry 1).
Hundreds: 4 + 2 + 1 = 7, which matches.
428 + 277 = 705. Check by inverse: 705 - 277 = 428.

For 6a x 7 = 4b9: the units digit of 7 x (units digit) is 9, which needs 7 x 7 = 49, so a = 7. Then 67 x 7 = 469, so b = 6.

Common errors

• Treating 1 as a prime, or forgetting that 2 is prime.
• Stopping a factor tree at a composite number such as 9.
• Writing 2³ x 3² as 6⁵: powers of different bases do not combine.
• Using the HCF when the question asks when events coincide (LCM), or the other way round.
• Doing all multiplications before all divisions, which gives 4 instead of 36 for 48 ÷ 4 x 3.
• Reading -8 - (-15) as -23.
• Reversing the steps of a number puzzle in the original order instead of the opposite order.
• Forgetting to carry in a missing-digit sum, then not checking the answer by the inverse.

Official syllabus

OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026), published by OCR (Cambridge OCR): sections 1.01a, 1.02a to 1.02c, 1.03a and 1.04a.