Introduction
Understanding which of the following is not a multiple of 12 helps students master divisibility rules and strengthen number sense, making it a fundamental concept in arithmetic. This article guides you through a clear process, explains the underlying mathematics, and answers common questions so you can confidently identify non‑multiples of 12 in any list.
Steps
To determine which of the following is not a multiple of 12, follow these systematic steps. Each step is presented as a sub‑heading for easy reference Not complicated — just consistent..
1. List the numbers to test
Create a simple list of the numbers you need to evaluate.
- Write each number on its own line or separate them with commas.
- Ensure the list is complete; missing a number can lead to an incorrect answer.
2. Apply the divisibility rule for 12
A number is a multiple of 12 if it is divisible by both 3 and 4.
- Divisible by 3: sum of its digits must be a multiple of 3.
- Divisible by 4: the number formed by its last two digits must be a multiple of 4.
If a number satisfies both conditions, it is a multiple of 12; otherwise, it is not That's the part that actually makes a difference. Practical, not theoretical..
3. Perform the calculations
For each number in your list:
- Check the sum of digits – add them together.
- Check the last two digits – see if the resulting two‑digit number is divisible by 4.
Record the result (multiple or not) in a table for clarity.
4. Identify the non‑multiple
After testing all numbers, the one that fails either the 3 test or the 4 test (or both) is which of the following is not a multiple of 12. Highlight this number for emphasis Simple, but easy to overlook..
5. Verify your answer
Divide the identified number by 12 directly. If the quotient is not an integer, your identification is correct. This final check eliminates any doubt.
Scientific Explanation
What defines a multiple of 12?
A multiple of 12 is any integer that can be expressed as (12 \times n) where (n) is an integer. Put another way, the number can be divided evenly by 12 without leaving a remainder.
The role of 3 and 4
Since (12 = 3 \times 4) and 3 and 4 are coprime (they share no common factors other than 1), a number is divisible by 12 iff it is divisible by both 3 and 4. This is why the two‑step test works:
- Divisibility by 3: The sum of digits rule ensures the number contains a factor of 3.
- Divisibility by 4: Examining the last two digits checks for a factor of 4, because any integer can be written as (100a + b) where (b) is the last two digits; 100 is itself a multiple of 4, so only (b) matters.
Why the rule is reliable
Mathematically, if a number (x) satisfies (x \equiv 0 \pmod{3}) and (x \equiv 0 \pmod{4}), then (x \equiv 0 \pmod{12}) because 3 and 4 are relatively prime. This is a direct application of the Chinese Remainder Theorem.
Practical examples
- 36: digits sum = 3 + 6 = 9 (multiple of 3); last two digits = 36 (multiple of 4) → multiple of 12.
- 45: digits sum = 4 + 5 = 9 (multiple of 3); last two digits = 45 (not multiple of 4
5. Worked‑out example set
Suppose the list you are given is
[ {48,; 72,; 84,; 96,; 108,; 120,; 132,; 144,; 156,; 168} ]
Apply the two‑step test to each entry Surprisingly effective..
| Number | Sum of digits | Divisible by 3? | Last two digits | Divisible by 4? | Multiple of 12?
And yeah — that's actually more nuanced than it sounds Nothing fancy..
All ten numbers pass both checks, so every one of them is a multiple of 12.
Now imagine the list contains an extra entry, 157. Running the same test:
- Sum of digits = 1 + 5 + 7 = 13 → not a multiple of 3.
- Last two digits = 57 → not a multiple of 4.
Because it fails the 3‑test (and the 4‑test as well), 157 is the only number that is not a multiple of 12. A quick division confirms the result:
[ 157 \div 12 = 13\text{ remainder }1, ]
so the quotient is not an integer.
6. Common pitfalls and how to avoid them
| Pitfall | Why it happens | Remedy |
|---|---|---|
| Only checking one rule (e. | Always perform both checks; a number must satisfy both conditions. Worth adding: , just the 3‑test) | You might mistakenly label a number like 45 as a multiple of 12. But |
| Arithmetic slip in digit sum | Adding digits mentally can produce a simple error that flips the result. Worth adding: | |
| Assuming “multiple of 12” means “multiple of 6” | 6 = 2 × 3, missing the factor of 4. | |
| Mis‑reading the last two digits | Forgetting leading zeros (e.Day to day, | |
| Relying on a calculator for division | Over‑reliance can mask a conceptual misunderstanding. g.g., “08” in 108) can lead you to think the number is 8, which is still divisible by 4, but the habit of dropping the zero can cause errors with numbers like 104 (last two digits “04”). | Use the divisibility rules first; the calculator is a sanity check, not the primary method. |
7. Extending the method
The same two‑rule approach works for any composite number whose prime factors are pairwise coprime. For instance:
- Multiple of 30 → must be divisible by 2, 3, and 5.
- Multiple of 18 → must be divisible by 2 and 9 (or by 3 and 6).
When the factors share common divisors, you can simplify the test by removing the redundancy (e.g., for 24 = 3 × 8, testing for 3 and 8 suffices because 8 already includes the factor 2) That's the part that actually makes a difference..
8. Quick‑reference cheat sheet
| Target multiple | Divisibility tests needed |
|---|---|
| 12 | 3 + 4 (sum of digits, last two digits) |
| 15 | 3 + 5 (sum of digits, last digit 0 or 5) |
| 20 | 4 + 5 (last two digits, last digit 0) |
| 28 | 4 + 7 (last two digits, double‑last‑digit rule) |
| 36 | 4 + 9 (last two digits, sum of digits multiple of 9) |
Not obvious, but once you see it — you'll see it everywhere Most people skip this — try not to..
Keep this table handy; it reduces mental load when you encounter a new problem.
Conclusion
Identifying the lone non‑multiple of 12 in a list is a straightforward exercise once you internalise the dual‑rule test: verify divisibility by 3 via the digit‑sum, and by 4 via the final two digits. Because 3 and 4 share no common factors, satisfying both guarantees a clean division by 12, a fact underpinned by elementary modular arithmetic and the Chinese Remainder Theorem.
By methodically applying these checks, recording results, and performing a final direct division for confirmation, you can pinpoint the outlier with confidence and avoid common arithmetic slips. The same logical framework extends to many other composite numbers, making it a versatile tool in both classroom settings and everyday numeric reasoning.
Easier said than done, but still worth knowing Most people skip this — try not to..
So, the next time you’re asked, “Which of the following is not a multiple of 12?”, you’ll know exactly how to answer—quickly, accurately, and with mathematical rigor.
