Unit 2 · Number Theory
Whole Numbers — LCM and Divisibility Count
Lesson 1 of 15
Welcome to Number Theory — and let's begin with one of its most practical tools: finding numbers that must be divisible by two different values at the same time. This is where the concept of the Lowest Common Multiple (LCM) earns its keep.
Divisibility and LCM questions feature in virtually every OC MR paper. They are usually placed early in the test and are among the most reliably scorable questions for students who know the LCM shortcut.
The examiner is checking whether you can translate "divisible by both A and B" into the single condition "divisible by LCM(A, B)", and then efficiently count how many such multiples fall within a given range — without listing every single number.
You are given a range (e.g. 1 to 100) and two divisors. Your task is to count how many numbers in that range are divisible by both. The trap is trying to check each number individually — that is too slow and error-prone.
Best approach: Calculate LCM(A, B) first — for coprime values (no shared factors) this is simply A × B. Then divide the upper bound of the range by the LCM and take the whole-number part of the result. That quotient is your answer. Always double-check by listing the last two or three multiples.
Question
How many whole numbers from 1 to 100 are divisible by both 3 and 5?
Select an answer to see the explanation.
Give Your Child the Best Chance at OC Entry
Join NSW families preparing their children for the Opportunity Class Placement Test with the most realistic online OC practice tests available. First tests free—no credit card required.
Claim Your Free OC Practice Tests