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

These notes condense sections 1.01 to 1.04 of the OCR GCSE (9-1) Mathematics J560 specification, Version 2.1 (August 2026): four rules with integers, whole number theory, priority of operations and inverse operations. All of it is Foundation and Higher content, with no Higher tier only statements in this unit. Full explanations are in the study guide, and exam-style questions are in the practice set.

Course pages: OCR GCSE Mathematics hub and the printable checklist. To find out which of these skills needs work, try a free 10-minute diagnostic.

1.01a Four rules

Operation	Rule
Subtracting a negative	Same as adding: 5 - (-3) = 8
Adding a negative	Same as subtracting: 5 + (-3) = 2
Multiply or divide, same signs	Positive answer
Multiply or divide, different signs	Negative answer
Many factors	Even number of negatives gives positive

Long multiplication in steps

1. Split the second number into tens and units.
2. Multiply by each part.
3. Add the partial products.

Division in steps

1. Subtract chunks of the divisor (100 lots, 10 lots, units).
2. Keep a running remainder.
3. Add up the chunks to get the quotient.

Small reminder: 6 x 154 = 600 + 300 + 24 = 924, so 924 ÷ 6 = 154.

1.02a Terms to know

Term	Quick definition
Factor (divisor)	Divides exactly into the number
Multiple	In the number’s times table
Common factor / multiple	Shared by two or more numbers
Prime	Exactly two factors
Square / cube	n x n / n x n x n
Root	Reverses a power: √49 = 7, ∛64 = 4
Place value	Each place left is worth ten times more

Place value reminder: in 805 126, the 5 is worth 5 000 and the 8 is worth 800 000.

Useful squares to 15² (225) and cubes to 5³ (125), plus 10³ = 1 000, speed up everything else in this unit.

1.02b Primes and prime factorisation

Primes under 20: 2, 3, 5, 7, 11, 13, 17, 19.

Foundation tier also: any prime. Test each prime up to the square root of the number. If none divides it, it is prime.

Factorisation in steps

1. Divide by the smallest prime that fits.
2. Keep dividing the quotient until you reach 1.
3. Collect repeats into powers.
4. Multiply back to check.

150 ÷ 2 = 75,  75 ÷ 3 = 25,  25 ÷ 5 = 5,  5 ÷ 5 = 1
150 = 2 x 3 x 5²        check: 2 x 3 x 25 = 150

Uniqueness: every whole number above 1 has one prime factorisation, apart from the order of the factors. Different factor trees always give the same end result.

1.02c HCF and LCM

	HCF	LCM
Meaning	Largest number dividing both	Smallest number both divide into
By listing	Factors in common, take the largest	Multiples in common, take the smallest
Foundation tier also: from primes	Shared primes, lowest powers	All primes, highest powers
Typical wording	Largest, greatest, equal groups, no remainder	Next time together, smallest, repeat every

72 = 2³ x 3²
60 = 2² x 3 x 5
HCF = 2² x 3 = 12
LCM = 2³ x 3² x 5 = 360
Check: 12 x 360 = 4 320 = 72 x 60

1.03a Priority of operations

1. Brackets first, including fraction bars and the line of a root sign.
2. Next powers, roots and reciprocals.
3. Then multiplication and division, reading from the left.
4. Last, addition and subtraction, again from the left.

2 x (3 + 4)² - 5 x 6 = 2 x 49 - 30 = 98 - 30 = 68
20 - 12 ÷ 4 + 2     = 20 - 3 + 2   = 19

If you need a value to be worked out first, put it in brackets, or write the fraction with a bar.

1.04a Inverse operations

Operation	Inverse
Add	Subtract
Multiply	Divide
Square	Square root
Cube	Cube root

Number puzzle in steps

1. Write the steps in the order given.
2. Reverse the order.
3. Replace each step with its inverse.
4. Check by running the original forward.

Simplifying

391 - 199 = 391 + 1 - 200 = 192
17 x 25   = 17 x 100 ÷ 4 = 1 700 ÷ 4 = 425

Missing digits: work column by column from the units, tracking carries, or use the inverse operation (for a + b = c, a missing b is c - a).

Divisibility tests

These save time when factorising and when testing for primes.

Divisible by	Test
2	Last digit is even
3	Digit sum is a multiple of 3
4	Last two digits form a multiple of 4
5	Last digit is 0 or 5
9	Digit sum is a multiple of 9
10	Last digit is 0

Example: 7 344 has digit sum 7 + 3 + 4 + 4 = 18, so it is divisible by 9. Its last two digits, 44, are a multiple of 4, so it is divisible by 4 as well. Check: 7 344 ÷ 9 = 816.

Worked reminders

Mixed signs. (-4) x (-9) + (-20) ÷ 5 = 36 + (-4) = 36 - 4 = 32. Do the multiplication and the division first, then combine.

A longer factorisation. Divide 2 520 down by the smallest prime each time.

2 520 ÷ 2 = 1 260
1 260 ÷ 2 =   630
  630 ÷ 2 =   315
  315 ÷ 3 =   105
  105 ÷ 3 =    35
   35 ÷ 5 =     7
2 520 = 2³ x 3² x 5 x 7

Listing for HCF and LCM. For 18 and 27, the factors of 18 are 1, 2, 3, 6, 9, 18 and the factors of 27 are 1, 3, 9, 27. The common factors are 1, 3 and 9, so the HCF is 9. The multiples of 27 are 27, 54, 81 and the multiples of 18 are 18, 36, 54, so the LCM is 54.

Roots in an expression. 5 + 2 x (√100 - 4)² = 5 + 2 x (10 - 4)² = 5 + 2 x 36 = 5 + 72 = 77. The root is evaluated inside the bracket first, then the bracket is squared, then the multiplication, then the addition.

Reversing a power. I square a number and add 11 to get 92. Reverse: 92 - 11 = 81, and √81 = 9 for a positive number. Check: 9² = 81, 81 + 11 = 92.

Habits that catch errors

• Estimate first. Round each number and see if your answer is close. 18 x 23 is about 20 x 20 = 400, and 414 fits.
• Reverse every division by multiplying back.
• Reverse every subtraction by adding back.
• When a sum has several operations, rewrite the expression on every line, changing one thing at a time.
• Underline the question word: “highest” or “lowest”, “factor” or “multiple”, “sum” or “product”.
• For a prime factorisation, multiply your primes back together.

Must-know distinctions

• Factor or multiple. Factors are smaller than or equal to the number. Multiples are larger than or equal to it.
• HCF or LCM. HCF can never exceed the smaller number. LCM can never be smaller than the larger number.
• Prime or composite. A composite has more than two factors. The number 1 is neither.
• Power or product. 2³ means 2 x 2 x 2 = 8, not 2 x 3 = 6.
• Order of operations versus left to right. Equal-priority operations go left to right, but brackets and powers break that rule.

Quick self-test

1. Work out 73 - (-9) - 12.
2. Work out 18 x 23 by hand.
3. Work out 1 344 ÷ 16 by hand.
4. State whether 101 and 91 are prime. Give a factor pair for any that is not.
5. Write 180 as a product of prime factors, using power notation.
6. Find the HCF and LCM of 48 and 36.
7. Work out 3 x (2 + 5)² - 4 x 8.
8. Work out 30 - 18 ÷ 6 + 4.
9. Work out (5³ - 5) / (2 x 3).
10. Work out 17 x 25 by using inverse operations to simplify.
11. I treble a number, subtract 4, then halve the result. I get 13. What was my number?
12. Find the missing digits: 5a + b9 = 102.

Answers

1. 73 + 9 - 12 = 70
2. 18 x 20 = 360, 18 x 3 = 54, total 414
3. 16 x 84 = 1 344, so 84
4. 101 is prime (not divisible by 2, 3, 5 or 7, and 11 x 11 = 121 is already above it). 91 is not prime: 91 = 7 x 13.
5. 2² x 3² x 5
6. HCF 12, LCM 144
7. 3 x 49 - 32 = 115
8. 30 - 3 + 4 = 31
9. (125 - 5) / 6 = 120 / 6 = 20
10. 17 x 100 = 1 700, then 1 700 ÷ 4 = 425
11. Reverse: 13 x 2 = 26, 26 + 4 = 30, 30 ÷ 3 = 10. Check: 10 x 3 = 30, 30 - 4 = 26, 26 ÷ 2 = 13.
12. Units: a + 9 ends in 2, so a = 3 (carry 1). Tens: 5 + b + 1 = 10, so b = 4. a = 3, b = 4 (53 + 49 = 102).

Where marks are usually lost

• Calling 1 prime, or leaving a composite number such as 9 or 15 in a prime factorisation.
• Writing a prime factorisation with no powers when the question asks for power notation.
• Giving a list of common factors when the question asks for the highest one.
• Mixing up HCF and LCM in word problems. Ask “do the events coincide, or do I share out equally?”
• Doing a division before a multiplication that stands to its left.
• Dropping a minus sign when subtracting a negative.
• Giving an answer to a number puzzle without checking it forwards.
• Forgetting a carry in long multiplication or a missing-digit sum.
• Not showing working on a method question that must be done by hand.
• Writing only the bare answer where the question says “show that” or asks for working.

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.