CBSE Class 6 Maths Chapter 3: Playing With Numbers Notes

Glossary of Key Terms

Factor

A number that divides a given number exactly without leaving any remainder.

Multiple

A number obtained by multiplying a given number by any non-zero whole number.

Prime Number

A number greater than 1 that has exactly two factors: 1 and the number itself.

Composite Number

A number that has more than two factors.

Perfect Number

A number for which the sum of all its factors is equal to twice the number itself (e.g., 6 and 28).

Co-prime Numbers

Two numbers that have only 1 as their common factor.

Twin Primes

Two prime numbers whose difference is exactly 2 (e.g., 3 and 5, 11 and 13).

Understanding Factors and Multiples

In arithmetic, understanding how numbers interact is key. A factor of a number is an exact divisor of that number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 completely with zero remainder. The number of factors for any given number is always finite, and every factor is less than or equal to the number itself. On the other hand, a multiple is generated by multiplying the number by natural numbers (1, 2, 3, and so on). The multiples of 12 are 12, 24, 36, 48, and so forth. The number of multiples of a given number is infinite, and each multiple is greater than or equal to the number itself. Master these concepts quickly using YoLearn AI Flashcards to drill definitions and properties before your exam!

Quick Divisibility Test Rules

1. Divisibility by 2 — The number must end in an even digit: 0, 2, 4, 6, or 8.
2. Divisibility by 3 — The sum of all the digits in the number must be divisible by 3.
3. Divisibility by 4 — The number formed by the last two digits of the number must be divisible by 4.
4. Divisibility by 5 — The last digit of the number must be either 0 or 5.
5. Divisibility by 6 — The number must be divisible by both 2 and 3.
6. Divisibility by 8 — The number formed by the last three digits must be divisible by 8.
7. Divisibility by 9 — The sum of all the digits in the number must be divisible by 9.
8. Divisibility by 10 — The last digit of the number must be 0.
9. Divisibility by 11 — Find the difference between the sum of digits at odd places and the sum of digits at even places (starting from the right). If the difference is 0 or divisible by 11, the number is divisible by 11.

HCF (Highest Common Factor) vs LCM (Lowest Common Multiple)

Step-by-Step Solved Mini-Examples

• 1. Find prime factorization of 24: 24 = 2 x 2 x 2 x 3 = (2^3) x 3 2. Find prime factorization of 36: 36 = 2 x 2 x 3 x 3 = (2^2) x (3^2) 3. Find common prime factors with lowest powers: 2^2 and 3^1 4. Multiply them: 4 x 3 = 12. HCF of 24 and 36 is 12.
• 1. Find prime factorization of 12: 12 = 2 x 2 x 3 = (2^2) x 3 2. Find prime factorization of 18: 18 = 2 x 3 x 3 = 2 x (3^2) 3. Take the highest powers of all prime factors present: 2^2 and 3^2 4. Multiply them: 4 x 9 = 36. LCM of 12 and 18 is 36.
• 1. Sum of digits at odd places (from right): 4 + 8 + 1 = 13 2. Sum of digits at even places (from right): 2 + 0 = 2 3. Find the difference: 13 - 2 = 11 4. Since 11 is divisible by 11, the number 10,824 is divisible by 11.

Must remember

• The number 1 is unique; it is neither prime nor composite.
• 2 is the smallest prime number and the only even prime number. All other prime numbers are odd.
• Every number is a multiple of itself, and every number is a factor of itself.
• 1 is a factor of every number.
• The number of factors of a finite number is finite, whereas the number of its multiples is infinite.
• If two numbers are co-prime, their HCF is 1, and their LCM is equal to their product.
• For any two given numbers, HCF x LCM = Product of the two numbers.
• Any two consecutive natural numbers (like 14 and 15) are always co-prime.

CBSE Exam Traps & Cues

• The '1' Confusion: Students often lose marks by writing 1 as the smallest prime number. Remember, prime numbers must have exactly two distinct factors. 1 has only one factor (itself), so it is not prime.
• Divisibility of 11 Direction: When calculating odd and even place sums for divisibility by 11, always count systematically from right-to-left. Mixing up the order can lead to arithmetic errors.
• Perfect Number Definition: A perfect number is NOT a perfect square. Do not confuse 36 (perfect square) with 6 or 28 (perfect numbers). Write the exact definition: 'a number equal to the sum of all its proper factors'.

Quick Revision Check

• What is the smallest perfect number? Explain why. The smallest perfect number is 6. Its factors are 1, 2, 3, and 6. The sum of its factors (1 + 2 + 3 + 6) is 12, which is equal to twice the number (2 x 6).
• Express 44 as the sum of two odd primes. 44 can be expressed as 3 + 41 or 13 + 31, where both numbers in the pair are odd prime numbers.
• What is the HCF and LCM of two co-prime numbers like 8 and 15? Since 8 and 15 are co-prime, their only common factor is 1, so their HCF is 1. Their LCM is the product of the numbers: 8 x 15 = 120.
• List all the prime numbers less than 20. The prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, and 19.

Frequently Asked Questions