NCERT Solutions Class 6 Maths Chapter 3 Exercise 3.7: Playing with Numbers

How to Decide Between HCF and LCM in Word Problems

When reading a word problem, ask yourself: 'Am I looking for a number that is smaller or larger than the given numbers?' If you need to split or divide things into smaller, equal parts or find a common measuring unit that fits exactly, you are looking for a common factor. Since you want the largest such unit, you find the Highest Common Factor (HCF). Look for keywords like maximum, greatest, largest, or longest.

On the other hand, if you are looking for a larger number that represents multiple cycles, repetitions, or events happening together in the future, you are looking for a common multiple. Since you want the earliest or smallest point where they meet, you find the Lowest Common Multiple (LCM). Look for keywords like minimum, least, smallest, together, or intervals.

Step-by-Step Guide to Solve Exercise 3.7 Problems

1. Read and Identify Keywords — Scan the question for keywords like 'maximum' or 'minimum' to decide if you need HCF or LCM.
2. Perform Prime Factorization — Find prime factors of each number using factor trees or continuous division.
3. Calculate HCF or LCM — For HCF, multiply the common prime factors. For LCM, multiply the highest powers of all prime factors.
4. State the Final Answer with Units — Conclude your answer with proper physical units such as liters, meters, seconds, or hours.

Fully Solved Examples from Exercise 3.7

• Example 1 (Based on Q1): Two tankers contain 850 liters and 680 liters of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene out of both tankers in exact number of times. Solution: Since we need the maximum capacity, we must find the HCF of 850 and 680. 1. Find prime factorization of 850: $850 = 2 \times 5 \times 5 \times 17$ 2. Find prime factorization of 680: $680 = 2 \times 2 \times 2 \times 5 \times 17$ 3. Identify the common prime factors: $2$, $5$, and $17$. 4. Multiply common factors: $\text{HCF} = 2 \times 5 \times 17 = 170$. Therefore, the maximum capacity of the container is 170 liters.
• Example 2 (Based on Q3): The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly. Solution: To find the longest tape, we must find the HCF of 825, 675, and 450. 1. Prime factorization of 825: $825 = 3 \times 5 \times 5 \times 11$ 2. Prime factorization of 675: $675 = 3 \times 3 \times 3 \times 5 \times 5$ 3. Prime factorization of 450: $450 = 2 \times 3 \times 3 \times 5 \times 5$ 4. Identify common factors: $3$ and $5 \times 5$ (which is $5^2$). 5. Calculate HCF: $\text{HCF} = 3 \times 5 \times 5 = 75$. Therefore, the longest tape required is 75 cm.
• Example 3 (Based on Q6): Three traffic lights change after every 48 seconds, 72 seconds, and 108 seconds. If they change simultaneously at 7 a.m., at what time will they change simultaneously again? Solution: Since the lights change at intervals and will meet again, we need to find the LCM of 48, 72, and 108. 1. Prime factors of 48: $2^4 \times 3$ 2. Prime factors of 72: $2^3 \times 3^2$ 3. Prime factors of 108: $2^2 \times 3^3$ 4. LCM takes the highest power of each prime factor involved ($2$ and $3$): $\text{LCM} = 2^4 \times 3^3 = 16 \times 27 = 432 \text{ seconds}$ 5. Convert 432 seconds into minutes and seconds: $432 \text{ seconds} = 7 \text{ minutes and } 12 \text{ seconds}$ Therefore, the lights will change simultaneously again at 7:07:12 a.m.

YoLearn Exam Tips & Pitfalls to Avoid

1. Don't rush on keywords: While 'maximum' usually means HCF and 'minimum' means LCM, check the actual physical context of the problem. If you need to make things smaller, think factors. If you are building up to a larger quantity, think multiples.
2. Units mismatch: Ensure all quantities are in the same units before starting calculations. For example, convert meters to centimeters if the problem mixes them.
3. Calculation check: Double-check your prime factorization. A simple division mistake here can ruin the entire calculation. Always multiply your factors back to see if they equal the original number.

Practice Questions with Solutions

• Q: Find the least number which when divided by 12, 16, 24, and 36 leaves a remainder of 7 in each case. A: Step 1: Since we need the least number divided by these numbers, we first find the LCM of 12, 16, 24, and 36. Step 2: Find LCM of (12, 16, 24, 36) using the division method: - Divide by 2: (6, 8, 12, 18) - Divide by 2: (3, 4, 6, 9) - Divide by 2: (3, 2, 3, 9) - Divide by 2: (3, 1, 3, 9) - Divide by 3: (1, 1, 1, 3) - Divide by 3: (1, 1, 1, 1) LCM = $2 \times 2 \times 2 \times 2 \times 3 \times 3 = 16 \times 9 = 144$. Step 3: Since it must leave a remainder of 7 in each case, add 7 to the LCM: $144 + 7 = 151$. Final answer: The least number is 151.
• Q: Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps? A: Step 1: The keyword 'minimum distance' tells us we need to find the LCM of 63, 70, and 77. Step 2: Perform prime factorization: - $63 = 3 \times 3 \times 7 = 3^2 \times 7$ - $70 = 2 \times 5 \times 7$ - $77 = 7 \times 11$ Step 3: Calculate the LCM by multiplying the highest powers of all prime factors ($2, 3^2, 5, 7, 11$): LCM = $2 \times 3^2 \times 5 \times 7 \times 11 = 2 \times 9 \times 5 \times 7 \times 11 = 90 \times 77 = 6930$. Final answer: The minimum distance each should cover is 6930 cm.
• Q: Find the smallest 4-digit number which is divisible by 18, 24 and 32. A: Step 1: To find a number divisible by 18, 24, and 32, we first find their LCM. Step 2: Prime factors: - $18 = 2 \times 3^2$ - $24 = 2^3 \times 3$ - $32 = 2^5$ LCM = $2^5 \times 3^2 = 32 \times 9 = 288$. Step 3: The smallest 4-digit number is 1000. Now, we find the multiple of 288 closest to but greater than or equal to 1000. Divide 1000 by 288: $1000 \div 288 \approx 3.47$. Multiply 288 by the next whole number (4): $288 \times 4 = 1152$. Final answer: The smallest 4-digit number divisible by 18, 24, and 32 is 1152.
• Q: Determine the greatest 3-digit number exactly divisible by 8, 10 and 12. A: Step 1: Find the LCM of 8, 10, and 12. - $8 = 2^3$ - $10 = 2 \times 5$ - $12 = 2^2 \times 3$ LCM = $2^3 \times 3 \times 5 = 8 \times 15 = 120$. Step 2: The greatest 3-digit number is 999. We need to find the largest multiple of 120 below 999. Step 3: Divide 999 by 120: $999 \div 120 = 8$ with a remainder of 39. Step 4: Subtract the remainder from 999 to get the exact multiple: $999 - 39 = 960$. Final answer: The greatest 3-digit number exactly divisible by 8, 10, and 12 is 960.

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