Playing With Numbers Class 6 NCERT CBSE Chapter 3 Guide

Understanding Factors and Multiples

Let's start with a fun visualization on our virtual sketchpad. Imagine you have 12 chocolates. In how many ways can you arrange them in equal rows?

• 1 row of 12 chocolates ($1 \times 12$)
• 2 rows of 6 chocolates ($2 \times 6$)
• 3 rows of 4 chocolates ($3 \times 4$)
• 4 rows of 3 chocolates ($4 \times 3$)
• 6 rows of 2 chocolates ($6 \times 2$)
• 12 rows of 1 chocolate ($12 \times 1$)

The numbers $1, 2, 3, 4, 6,$ and $12$ are called the factors of 12 because they divide 12 completely without leaving any remainder. Remember, a factor of a number is an exact divisor of that number. Every number has a finite number of factors, and $1$ is a factor of every number.

Now, what happens if we count in steps of 12? $12, 24, 36, 48, 60...$ These are the multiples of 12. Think of multiples as the products you get when you multiply 12 by natural numbers ($1, 2, 3...$). Unlike factors, the number of multiples of any given number is infinite!

Fast-Track Divisibility Rules

1. Divisibility by 2 and 5 — Look at the last digit. If the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2. If the last digit is 0 or 5, it is divisible by 5.
2. Divisibility by 3 and 9 — Add up all the digits of the number. If the sum is divisible by 3, the entire number is divisible by 3. If the sum is divisible by 9, the number is divisible by 9.
3. Divisibility by 6 — A number is divisible by 6 if it is divisible by both 2 and 3 simultaneously.
4. Divisibility by 4 and 8 — If the last two digits form a number divisible by 4, the whole number is divisible by 4. If the last three digits form a number divisible by 8, the whole number is divisible by 8.
5. 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 this difference is 0 or divisible by 11, the number is divisible by 11.

Finding HCF and LCM Step-by-Step

• Example 1: Find the Highest Common Factor (HCF) of 20, 28, and 36 - Step 1: Write down the prime factorization of each number: - $20 = 2 \times 2 \times 5 = 2^2 \times 5$ - $28 = 2 \times 2 \times 7 = 2^2 \times 7$ - $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$ - Step 2: Identify the common prime factors. The only common prime factor is 2. - Step 3: Find the lowest power of the common prime factors. The lowest power of 2 is $2^2$. - Step 4: Multiply these common prime factors. Here, it is just $2 \times 2 = 4$. - HCF is 4.
• Example 2: Find the Lowest Common Multiple (LCM) of 12, 16, and 24 using the Common Division Method - Step 1: Arrange the numbers in a row, separated by commas: 12, 16, 24 - Step 2: Divide by the smallest prime number (2) that divides at least one of them: - $12 \div 2 = 6$ - $16 \div 2 = 8$ - $24 \div 2 = 12$ - Row becomes: 6, 8, 12 - Step 3: Keep dividing by 2: - Row becomes: 3, 4, 6 - Divide by 2 again: 3, 2, 3 (3 is carried forward because it is not divisible by 2) - Divide by 2 again: 3, 1, 3 - Step 4: Divide by the next prime number (3): - Row becomes: 1, 1, 1 - Step 5: Multiply all the prime divisors used: $2 \times 2 \times 2 \times 2 \times 3 = 48$. - LCM is 48.

Pro-Tips for Your Exams

Avoid these common exam traps:

1. Is 1 Prime or Composite? Neither! 1 is a unique number because it has only one factor (itself). Prime numbers must have exactly two distinct factors (1 and the number itself).
2. HCF vs. LCM Confusion: Remember, HCF stands for Highest Common Factor but the final answer is always smaller than or equal to the given numbers. LCM stands for Lowest Common Multiple but the final answer is always larger than or equal to the given numbers.
3. Co-prime Numbers: Two numbers are co-prime if their only common factor is 1. They do not have to be prime numbers themselves! For example, 8 and 9 are co-prime, even though both are composite numbers.

Practice Questions with Solutions

• Q: Find all the factors of 45. A: Step 1: Find pairs of numbers that multiply together to give 45: - $1 \times 45 = 45$ - $3 \times 15 = 45$ - $5 \times 9 = 45$ Step 2: Collect all the numbers involved in these multiplications in ascending order. Final answer: The factors of 45 are 1, 3, 5, 9, 15, and 45.
• Q: Check if the number 5382 is divisible by 9. A: Step 1: Find the sum of all digits of 5382. - Sum = $5 + 3 + 8 + 2 = 18$ Step 2: Test if the sum is divisible by 9. - $18 \div 9 = 2$ (with 0 remainder). Step 3: State the final conclusion based on the divisibility rule of 9. Final answer: Since the sum of digits (18) is divisible by 9, the number 5382 is divisible by 9.
• Q: Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container that can measure the petrol of either tanker in exact number of times. A: Step 1: Identify that 'maximum capacity' requires finding the Highest Common Factor (HCF) of 850 and 680. Step 2: Find the prime factorization of 850: - $850 = 2 \times 5 \times 5 \times 17 = 2 \times 5^2 \times 17$ Step 3: Find the prime factorization of 680: - $680 = 2 \times 2 \times 2 \times 5 \times 17 = 2^3 \times 5 \times 17$ Step 4: Determine the common prime factors with their lowest power: - Common factors are 2, 5, and 17. - HCF = $2 \times 5 \times 17 = 170$. Final answer: The maximum capacity of the container must be 170 litres.
• Q: Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many hours will they toll together next? A: Step 1: Identify that the next time they toll together is determined by finding the Lowest Common Multiple (LCM) of 9, 12, and 15. Step 2: Use prime factorization to find the LCM: - $9 = 3^2$ - $12 = 2^2 \times 3$ - $15 = 3 \times 5$ Step 3: Take the highest power of each prime factor involved: - Highest power of 2 is $2^2$ - Highest power of 3 is $3^2$ - Highest power of 5 is $5^1$ - LCM = $2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$ minutes. Step 4: Convert 180 minutes into hours: - $180 \div 60 = 3$ hours. Final answer: They will toll together next after 3 hours.

Frequently Asked Questions