Technique explanation
Teaching Explanation
Divisibility by a large composite number, such as 15 or 18, can be determined without performing the full division. By breaking the divisor into its factors, we can apply simpler rules sequentially. For a number to be divisible by 15, it must satisfy the divisibility rules for both 3 and 5. If it fails even one of these "sub-tests," it cannot be divisible by the original composite number. This reasoning allows for rapid verification of results and provides a deeper understanding of how numbers are structured through their prime factors.
Method Condition
Applies when the divisor is a composite number with known factor rules.
Standard Fallback
Direct division.
Misconception: Using any factors (like 2 and 6 for 12) without considering if they are coprime (sharing no common factors).
Correction: For the most reliable results, decompose the divisor into factors that do not share common parts, or use the specific prime factor rules you know best.
Worked examples
| Step | Example 1 | Example 2 |
|---|---|---|
| 1. Factors | $15 = 3 \times 5$. | $12 = 3 \times 4$. |
| 2. Test 1 | Ends in 0; divisible by 5. Yes. | Last two digits (56) are div by 4. Yes. |
| 3. Test 2 | Digit sum $1+2+3+0=6$; div by 3. Yes. | Digit sum $4+5+6=15$; div by 3. Yes. |
| 4. Conclusion | Divisible by 15. | Divisible by 12. |
| 5. Check | $1,230 \div 15 = 82$. | $456 \div 12 = 38$. |
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