Divisibility Rules, Prime Numbers and Composite Numbers
Number theory forms the logical bedrock of advanced mathematical reasoning, providing mathematicians with powerful rules to analyze integer structures without performing exhaustive long division. In this introductory lesson of Grade 6 Mathematics, students investigate the universal divisibility rules for 2, 3, 4, 5, 6, 9, and 10. Understanding that divisibility by 2 and 5 depends strictly on the final units digit, while divisibility by 3 and 9 relies upon the sum of digits, unlocks rapid computational fluency. Learners also classify integers into prime numbers—possessing exactly two distinct positive factors—and composite numbers possessing three or more factors. By establishing that 2 is the unique even prime number and that 0 and 1 are neither prime nor composite, students construct a robust foundation for multi-digit prime factorization and advanced algebraic applications.
1. Universal Divisibility Rules Matrix
A whole number is divisible by another non-zero whole number if division results in a whole quotient with a remainder of zero:
| Divisor | Divisibility Test Condition | Illustrative Example | Mathematical Rationale |
|---|---|---|---|
| 2 | The ones digit is even (0, 2, 4, 6, 8) | 3,478 (ends in 8 → divisible) | Any multiple of 10 is even; thus only the units place matters. |
| 3 | The sum of all digits is a multiple of 3 | 4,521 → 4+5+2+1 = 12 (divisible) | Each power of 10 leaves a remainder of 1 when divided by 3 ($10^n \equiv 1 \pmod 3$). |
| 4 | The number formed by the last two digits is divisible by 4 | 7,524 → 24 ÷ 4 = 6 (divisible) | Since 100 is divisible by 4, all hundreds and thousands are automatically divisible by 4. |
| 5 | The ones digit is either 0 or 5 | 1,895 (ends in 5 → divisible) | Multiples of 5 always alternate between 0 and 5 in the units position. |
| 6 | The number is divisible by BOTH 2 and 3 | 642 (ends in 2 and 6+4+2=12 → divisible) | Since 6 = 2 × 3 and gcd(2,3) = 1, composite divisibility holds. |
| 9 | The sum of all digits is a multiple of 9 | 8,712 → 8+7+1+2 = 18 (divisible) | Each power of 10 leaves a remainder of 1 when divided by 9 ($10^n \equiv 1 \pmod 9$). |
| 10 | The ones digit is 0 | 9,450 (ends in 0 → divisible) | Base-10 place value ensures that integers ending in 0 contain factor 10. |
2. Prime vs. Composite Numbers
In the set of natural numbers greater than 1 ($\mathbb{N}_{>1}$), every integer can be classified into one of two fundamental categories:
Prime Numbers
A whole number greater than 1 that has exactly two distinct positive factors: 1 and itself.
- Primes under 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
- Special Fact: 2 is the smallest prime and the only even prime number.
Composite Numbers
A whole number greater than 1 that has more than two positive factors.
- Examples: 4 (factors: 1, 2, 4), 6 (1, 2, 3, 6), 9 (1, 3, 9), 12 (1, 2, 3, 4, 6, 12).
- Every composite number can be uniquely factored into primes.
Critical Axiom: Zero and One
The number 1 is neither prime nor composite because it has only one positive factor (itself). The number 0 is also neither prime nor composite because it has infinitely many factors ($0 × k = 0$).
Practical Application & Problem Solving Workshop
Activity 1: Multi-Divisibility Verification
Examine the number 4,320 and determine which of the divisors (2, 3, 4, 5, 6, 9, 10) divide it evenly.
- By 2: Yes, ends in 0 (even).
- By 3: Yes, 4+3+2+0 = 9 (multiple of 3).
- By 4: Yes, last two digits 20 is divisible by 4 (20 ÷ 4 = 5).
- By 5: Yes, ends in 0.
- By 6: Yes, divisible by both 2 and 3.
- By 9: Yes, sum of digits 9 is divisible by 9.
- By 10: Yes, ends in 0.
Conclusion: 4,320 is divisible by all seven numbers (2, 3, 4, 5, 6, 9, 10).
Activity 2: Prime Identification
Is 91 a prime number? Justify your reasoning mathematically.
Tap any card to reveal its meaning, or tap the audio icon to listen:
Click on an option to check your answer and view the detailed explanation: