What is the least common multiple

2018 Handbook

2018 Test B Math · Question 14 of 57

Also in 2025 Practice Test B Math, question 14.

Question

What is the least common multiple of 24, 6, and 18?

A36
B48
C72
D144

Show the answer

Correct answer: C

Since 18 and 24 are both multiples of 6, find the least common multiple of only 18 and 24.

Multiples of 18:

18, 36, 54, 72…

Multiples of 24:

24, 48, 72…

The least common multiple of 6, 18, and 24 is 72.

Question analysis

This number properties question asks for the least common multiple of three numbers: 24, 6 and 18. Students can list multiples or use prime factors. Each candidate has to be checked against all three numbers, and it must be the smallest number that works, not just any shared multiple.

NYS standards: 6.NS.4 common factors and multiples
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Least common multiples are grade 6 content, where New York's standard uses two numbers up to 12. This uses three larger numbers, but it is still a short factor task below typical grade 7 work.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Every choice must be checked minor
Something you must already know minor
Solution: 2 steps
Reading level: Question and choices: about grade 4, 8 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Finding a least common multiple is routine, and listing multiples of 24 reaches the answer by the third one. The wrong choices are built to catch incomplete checking: each passes the test for some of the numbers but fails for one, or passes for all and is simply not the smallest. Testing every candidate against all three numbers is the whole skill.

Every choice must be checked minor Choosing the answer
Each wrong choice passes part of the test: 36 is a multiple of 6 and 18, 48 of 6 and 24, and 144 of all three, so every candidate needs checking against all three numbers. (Embretson & Wetzel, 1987)
Something you must already know minor Working it out
Students must know what “least common multiple” means and see that 6 can be dropped, since it already divides both 24 and 18. (Anderson & Krathwohl, 2001)

Strategy: Note that 6 divides both 24 and 18, so only those two matter; list multiples of 24 and stop at the first one that 18 also divides.

Traps

Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds the least common multiple of 6 and 18 and stops, never checking that 36 is not a multiple of 24.
B An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: weak
Lists multiples of 24, stops at 48 and misjudges it as a multiple of 18, though 48 ÷ 18 is not a whole number.
D Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: moderate
Multiplies 24 by 6 to get 144, which all three numbers divide, overlooking that the question asks for the least such multiple.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Multiples of a number: $n \times 1,\ n \times 2,\ n \times 3,\ \ldots$; n is any whole number. Its multiples are what you get when you multiply n by 1, 2, 3 and so on. The least common multiple of some numbers is the smallest number that is a multiple of every one of them.

Understand the question
A multiple of a number is that number times 1, 2, 3, and so on. A common multiple is a multiple of every number in the list.
You need the least, or smallest, common multiple of 24, 6 and 18.
Drop the 6
6 goes evenly into 24, since $6 \times 4 = 24$ . It also goes into 18, since $6 \times 3 = 18$ .
So any multiple of 24 is already a multiple of 6. You only need to work with 24 and 18.
List the multiples of 24
Start with the bigger number, since its list grows faster. Check each multiple to see if 18 divides it evenly.
- $24 \times 1 = 24$ , and 24 ÷ 18 is not a whole number.
- $24 \times 2 = 48$ , and 48 ÷ 18 is not a whole number.
- $24 \times 3 = 72$ , and $72 \div 18 = 4$ . It works.
Give the answer
72 is the first multiple of 24 that 18 also divides. It is a multiple of 6 too, since $72 \div 6 = 12$ .
Watch out: 36 works for 6 and 18, but not for 24, so choice A is wrong. 144 works for all three, but it is not the smallest.
So the answer is C, 72.

Check your answer

Test 72 with all three numbers: $72 \div 24 = 3$ , $72 \div 6 = 12$ and $72 \div 18 = 4$ . All are whole numbers. The smaller choices, 36 and 48, each fail for one of the numbers.

The original DOE question

Question 71 as printed on page 155 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 155, question 71. Open page 155 of the handbook (PDF)