Unit 2
Number Theory
About this unit
Explore the properties of whole numbers — divisibility, factors, multiples, LCM, HCF, primes, and combinatorics. Build the number sense that underpins every other MR topic.
What types of questions will you face?
- 1Find how many numbers in a given range are divisible by two (or more) values simultaneously — requires computing their LCM first
- 2Solve two-digit number puzzles where clues involve the digit sum and what happens when the digits are reversed — set up two simultaneous equations
- 3Count how many numbers in a range satisfy a digit-based condition (e.g. digit sum equals 9) by listing pairs systematically
- 4Calculate the total number of handshakes, games, or connections when every pair in a group meets exactly once — use the formula n(n−1) ÷ 2
- 5Find the smallest (or largest) number satisfying multiple divisibility conditions by finding the LCM or HCF of the given values
Skills you will build
- Computing the LCM of two or more numbers to convert "divisible by both A and B" into a single clean condition
- Representing a two-digit number algebraically as 10a + b to write equations from digit-sum and reversal clues, then solving the system
- Counting digit pairs systematically — listing valid (tens, units) combinations without omissions or double-counting
- Applying the handshake formula n(n−1) ÷ 2 quickly to count unique pairwise meetings in any group
- Identifying factors, multiples, prime factorisations, and divisibility rules quickly to save time on screening questions
By the end of this unit, you will be able to
- Use LCM confidently to solve range-divisibility problems — translate two conditions into one and count multiples in one step
- Set up and solve a pair of simultaneous equations to crack any two-digit digit-sum puzzle
- Count integer solutions within a range that satisfy digit or divisibility constraints without missing any case
- Apply the handshake formula accurately and instantly without drawing diagrams or listing every pair
Difficulty profile
Questions in this unit range from Very Easy to Medium, with occasional Difficult combinatorics items appearing late in the test. LCM and digit-sum problems are generally Very Easy to Easy; handshake and path-counting questions rise to Medium or Difficult and separate the top scorers.
Exam tip: Number Theory
When a question says a number must be divisible by A and by B, your very first move is to find LCM(A, B). Never check each number in the range individually — it is too slow and prone to error. For digit puzzles, always label the tens digit a and the units digit b straight away, write both equations, and add or subtract to eliminate one variable.
Give Your Child the Best Chance at Selective Entry
Join NSW families preparing their children for the Selective Schools Placement Test with the most realistic online Selective practice tests available. First tests free—no credit card required.
Claim Your Free Selective Practice Tests