3. Finding Common Ground
Answers to all Figure it Out and in-text questions on HCF, LCM, prime factorisation and their properties. CBSE / NCERT Class 7 Ganita Prakash Part 2.
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3.1 The Greatest of All
The largest tile covers the floor with the fewest tiles.
Tiles needed = (12 ÷ 4) × (16 ÷ 4) = 3 × 4 = 12 tiles.
If the side could be a fraction of a foot, the answer is still 4 ft. A square that fits exactly must divide both 12 and 16 a whole number of times, and 4 ft is the largest such size.
The common factors are 1, 2, 3, 4, 6 and 12. The largest is 12 kg.
Bags = 84 ÷ 12 + 108 ÷ 12 = 7 + 9 = 16 bags.
(a) 14 and 30 → 2
(b) 7 and 11 → 1
(c) 30 and 50 → 10
(d) 28 and 42 → 14
Yes, it is their HCF. The jump size must divide both numbers, and we want the largest such number.
1200 = 2 × 2 × 2 × 2 × 3 × 5 × 5. The division method is easier, because we divide by one prime at a time.
14: Yes. 2 × 7 is a subpart. 840 = 14 × 60.
2 × 2 × 2 = 8: Yes. 840 has three 2s. 840 = 8 × 105.
3 × 3 × 3 = 27: No. 840 has only one 3.
(a) 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
(b) 105: 1, 3, 5, 7, 15, 21, 35, 105
(c) 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
(d) 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
(e) 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840
(a) 50 = 2 × 5 × 5, 60 = 2 × 2 × 3 × 5. Common factors: 1, 2, 5, 10. HCF = 10
(b) 140 = 2 × 2 × 5 × 7, 275 = 5 × 5 × 11. Common factors: 1, 5. HCF = 5
(c) 77 = 7 × 11, 725 = 5 × 5 × 29. Common factor: 1. HCF = 1
(d) 370 = 2 × 5 × 37, 592 = 2 × 2 × 2 × 2 × 37. Common factors: 1, 2, 37, 74. HCF = 74
(e) 81 = 3 × 3 × 3 × 3, 243 = 3 × 3 × 3 × 3 × 3. Common factors: 1, 3, 9, 27, 81. HCF = 81
1. Find the HCF.
(a) 24, 180 → 12
(b) 42, 75, 24 → 3
(c) 240, 378 → 6
(d) 400, 2500 → 100
(e) 300, 800 → 100
2. 72 = 6 × 12 and 144 = 8 × 18. Can we say they have no common factor other than 1?
No. 6, 12, 8 and 18 are not primes. We must break them into primes: 72 = 2 × 2 × 2 × 3 × 3 and 144 = 2 × 2 × 2 × 2 × 3 × 3. In fact, the HCF is 72.
3.2 Least, but not Last!
No. If a number is a common multiple, then 2 times it, 3 times it, and so on are also common multiples. So, there is no largest one.
(a) 4 and 6 → 12
(b) 7 and 11 → 77
(c) 14 and 30 → 210
(d) 15 and 55 → 165
Yes, it is always the LCM, because it is the first number that is a multiple of both.
Yes. 210 contains 2 × 7 and 5 × 7, so 210 = 14 × 15 = 35 × 6.
Extra 2s make a bigger common multiple. Five 2s are enough to contain the 2s of both numbers.
(a) 30, 72 → 360
(b) 36, 54 → 108
(c) 105, 195, 65 → 1365
(d) 222, 370 → 1110
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3.3 Patterns, Properties, and a Pretty Procedure!
Examples: 5 and 20, 8 and 24, 9 and 9. One number is a factor of the other.
(a) The other number can be any multiple of m, like 2m or 5m, or any factor of m.
(b) For 7k, the other number can be 14k, 21k, … or a factor of 7k such as k or 7.
1. General statements about the HCF
(a) Two consecutive even numbers: HCF = 2. They differ by 2, so a common factor must divide 2.
(b) Two consecutive odd numbers: HCF = 1. They differ by 2 and are odd.
(c) Two even numbers: the HCF is even (at least 2).
(d) Two consecutive numbers: HCF = 1. They differ by 1.
(e) Two co-prime numbers: HCF = 1 (by definition).
2. The LCM of 3 and 24 is 24.
(a) Other pairs: 4 and 12, 5 and 25, 6 and 30, 7 and 7.
(b) The LCM is one of the numbers when one number is a multiple of the other: n and kn have LCM kn.
3. General statements about the LCM
(a) Two multiples of 3: the LCM is a multiple of 3.
(b) Two consecutive even numbers 2n and 2n + 2: LCM = half their product, since their HCF is 2.
(c) Two consecutive numbers: LCM = their product.
(d) Two co-prime numbers: LCM = their product.
(a) 90 (b) 10 (c) 5 (d) 48
The HCF equals the common multiplier in (b) and (c). This happens when the other two numbers (38, 21 and 13, 20) are co-prime.
84 and 180: divide by 2, 2, 3 to get 7 and 15 (no common factor). HCF = 2 × 2 × 3 = 12. LCM = 12 × 7 × 15 = 1260.
300 and 150: HCF = 2 × 5 × 5 × 3 = 150, LCM = 150 × 2 × 1 = 300. 630 and 770: HCF = 70, LCM = 70 × 9 × 11 = 6930.
The LCM is the HCF times the leftover numbers. It contains the full prime factorisation of each number, and nothing extra.
Guna’s and Anshu’s methods work because dividing by a big common factor is the same as dividing by its primes one after another.
Try: (a) 90 and 150 → HCF 30, LCM 450. (b) 84 and 132 → HCF 12, LCM 924.
The product is a common multiple of the two numbers. The LCM is the lowest common multiple. So, the LCM is never greater than the product. They are equal when the HCF is 1.
| Numbers | Product | LCM | Multiplied by | HCF |
|---|---|---|---|---|
| 45, 105 | 4725 | 315 | 15 | 15 |
| 275, 352 | 96,800 | 8800 | 11 | 11 |
| 222, 370 | 82,140 | 1110 | 74 | 74 |
In each case, the multiplier is the HCF. So, HCF × LCM = product of the two numbers.
Reason: a prime that occurs in both numbers goes to the HCF (smaller count) and to the LCM (larger count). Together they have exactly as many of that prime as the product.
It does not hold for 3 numbers. Example: 2, 4, 8 have HCF 2 and LCM 8, so HCF × LCM = 16, but the product is 64.
1. When will the blue stars meet next?
In the first row, blue comes at positions 4, 10, 16, 22, … (every 6). In the second row, at 4, 8, 12, 16, … (every 4).
They meet at position 4 and next at position 16 (then every 12 positions).
2. (a) Is 5 × 7 × 11 × 11 a multiple of 5 × 7 × 7 × 11 × 2? (b) Is it a factor?
(a) No. It does not contain 7 × 7 or 2.
(b) No. It has two 11s, but the other has only one.
3. Find the HCF and LCM as prime factorisations.
(a) 3 × 3 × 5 × 7 × 7 and 2 × 2 × 3 × 7 × 11: HCF = 3 × 7, LCM = 2 × 2 × 3 × 3 × 5 × 7 × 7 × 11
(b) 45 = 3 × 3 × 5 and 36 = 2 × 2 × 3 × 3: HCF = 3 × 3, LCM = 2 × 2 × 3 × 3 × 5
4. Two numbers with HCF 1 and LCM 66
6 and 11. (Also 2 and 33, 3 and 22, 1 and 66.)
5. How many cows (less than 200)?
The number is a multiple of 3, 5 and 7, so a multiple of 105. Less than 200 gives 105 cows.
6. Which cubes pack the 12 × 18 × 36 box without gaps?
The side must divide 12, 18 and 36. So, (b) 6 cm, (d) 3 cm and (e) 2 cm.
7. Largest number that divides both 306 and 36
(c) 18 (it is the HCF).
8. Smallest number divisible by 3, 4, 5, 7 that leaves remainder 10 when divided by 11
It must be a multiple of 420. 420, 840, 1260, 1680 do not work. 2100 = 11 × 190 + 10 works.
9. No one got out for 6 and 9, but some got out for 10.
The number is a multiple of 6 and 9 (that is, of 18) but not of 10. So, (a) 72 or (e) 36.
10. The LCM of two different primes m and n can be:
The LCM is m × n. So, only (c) greater than both numbers is correct.
11. A dog chases a rabbit with a 150 ft head start (dog 9 ft, rabbit 7 ft per leap).
The dog gains 2 ft in each leap. 150 ÷ 2 = 75 leaps.
12. Smallest multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10
LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360.
13. Add 815, 120, 736, 1163 and 121.
Use the LCM of the denominators, 1260: 6721260 + 631260 + 2451260 + 2201260 + 601260 = 12601260 = 1.
At 1 digit per second, it takes 4,10,24,320 seconds. That is about 4,10,24,320 ÷ 86,400 ≈ 475 days without any break.
Each prime number has its own colour. The circle is split into as many parts as there are prime factors (counting repeats), and each part has the colour of that prime.
Example: 12 = 2 × 2 × 3 has three parts — two of the colour for 2 and one of the colour for 3. A prime page number has a full circle of one colour.
For 101 – 110: 101, 103, 107 and 109 are primes (one new colour each); 102 = 2 × 3 × 17; 104 = 2 × 2 × 2 × 13; 105 = 3 × 5 × 7; 106 = 2 × 53; 108 = 2 × 2 × 3 × 3 × 3; 110 = 2 × 5 × 11.