CBSE Class 7 Mathematics Finding Common Ground MCQs Set 11

Answer: (c) the highest of their common factors

Explanation:
1. Common factors are the numbers that divide both the given numbers.
2. The HCF is the greatest number in this list.
3. So the HCF is the highest of their common factors.

Teacher's Note:
1. The name itself says it: highest common factor.
2. The smallest common factor is always 1, so it tells us nothing.

Answer: (b) Greatest Common Divisor

Explanation:
1. A factor of a number is also called a divisor of it.
2. The HCF is the greatest number that divides both numbers.
3. So it is also called the Greatest Common Divisor (GCD).

Teacher's Note:
1. Factor and divisor mean the same here.
2. HCF and GCD are two names for one idea.

Answer: (d) 1, 2, 4

Explanation:
1. Factors of 12: 1, 2, 3, 4, 6, 12.
2. Factors of 16: 1, 2, 4, 8, 16.
3. Numbers in both lists: 1, 2, 4.
4. 3 does not divide 16, and 8 does not divide 12.

Teacher's Note:
1. List the factors of each number fully first.
2. Do not forget 1, which is a common factor of every pair.

Answer: (a) greater than 1 whose only factors are 1 and the number itself

Explanation:
1. A prime is greater than 1 and has exactly two factors: 1 and itself.
2. Option (b) fails: 9 is odd but 9 = 3 × 3 is not prime.
3. Option (c) fails: a prime has two factors, not three. So the answer is (a).

Teacher's Note:
1. 2 is a prime even though it is even.
2. 1 is not a prime because it has only one factor.

Answer: (a) Only Statement I is true

Explanation:
1. A prime cannot be split further, so its prime factorisation is just itself, e.g. 17 = 17. Statement I is true.
2. A prime must be greater than 1, so 1 is not a prime. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. 1 is neither prime nor composite.
2. A prime is its own prime factorisation.

Answer: (c) the lowest of their common multiples

Explanation:
1. Common multiples are numbers that are multiples of both the given numbers.
2. The LCM is the smallest number in this list.
3. So the LCM is the lowest of their common multiples; there is no highest one.

Teacher's Note:
1. LCM is about multiples, HCF is about factors.
2. Common multiples go on for ever, so only a lowest exists.

Answer: (d) 36 cm

Explanation:
1. The toran length must be a multiple of 9 and also a multiple of 12, so it is a common multiple.
2. 9 = 3 × 3 and 12 = 2 × 2 × 3.
3. LCM = 2 × 2 × 3 × 3 = 36, so the shortest toran is 36 cm.

Teacher's Note:
1. Joining strips end to end gives multiples of the strip length.
2. 108 is a common multiple but not the shortest.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. A common factor divides the smaller number too, so it cannot be bigger than the smaller number. A is true.
2. R states exactly this fact, and it is true.
3. R gives the reason for A, so the answer is (a).

Teacher's Note:
1. A factor never exceeds the number it divides.
2. The HCF can equal the smaller number, e.g. HCF(7, 42) = 7.

Answer: (d) (i)-(q), (ii)-(r), (iii)-(p)

Explanation:
1. 90 = 2 × 3 × 3 × 5, so (i)-(q).
2. 105 = 3 × 5 × 7, so (ii)-(r).
3. 112 = 2 × 2 × 2 × 2 × 7, so (iii)-(p).
4. The correct matching is (i)-(q), (ii)-(r), (iii)-(p).

Teacher's Note:
1. Multiply out each product in Column II to check.
2. Only 105 is odd, so its factorisation has no 2.

Answer: (c) their HCF is 1

Explanation:
1. Co-prime numbers have no common factor other than 1.
2. So their HCF is 1.
3. They need not be primes: 8 and 9 are co-prime, but neither is prime.

Teacher's Note:
1. Co-prime describes a pair, not a single number.
2. 8 and 9 is a good example to remember.

Answer: (b) 6 ft

Explanation:
1. The tile side must divide 30 ft and 18 ft exactly, so it is a common factor of 30 and 18.
2. Fewest tiles means the biggest tile, so we need the HCF.
3. 30 = 2 × 3 × 5 and 18 = 2 × 3 × 3, so HCF = 2 × 3 = 6. Each tile has side 6 ft.

Teacher's Note:
1. Biggest tile gives fewest tiles.
2. 9 ft does not fit along 30 ft, since 30 ÷ 9 is not whole.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. A is a true fact about prime factorisation.
2. R is true: 3 × 30 = 3 × 2 × 3 × 5 and 2 × 45 = 2 × 3 × 3 × 5.
3. But one example cannot explain why it always happens, so the answer is (b).

Teacher's Note:
1. An example supports a statement but does not prove it.
2. Always check whether R gives the reason or just a case.

Answer: (c) the maximum number of times it occurs in the two factorisations

Explanation:
1. A common multiple must contain the full prime factorisation of each number.
2. So each prime appears the maximum number of times it occurs in either factorisation.
3. Taking the minimum number of times gives the HCF instead.

Teacher's Note:
1. LCM: maximum; HCF: minimum.
2. Adding the counts together gives the product, not the LCM.

Answer: (b) 36

Explanation:
1. 144 = 2 × 2 × 2 × 2 × 3 × 3.
2. 180 = 2 × 2 × 3 × 3 × 5.
3. Common primes, minimum times: 2 × 2 × 3 × 3 = 36.

Teacher's Note:
1. Check: 144 ÷ 36 = 4 and 180 ÷ 36 = 5, and 4 and 5 share nothing.
2. 12 and 18 are common factors but not the highest.

Answer: (d) 120

Explanation:
1. 40 = 2 × 2 × 2 × 5.
2. 60 = 2 × 2 × 3 × 5.
3. Each prime the maximum number of times: 2 × 2 × 2 × 3 × 5 = 120.

Teacher's Note:
1. 20 is the HCF, not the LCM.
2. 2400 is the product, which is a common multiple but not the lowest.

Answer: (a) Both statements are true

Explanation:
1. HCF × LCM = product of the two numbers is a rule that always holds. Statement I is true.
2. The product is itself a common multiple, so the lowest common multiple cannot be bigger. Statement II is true.
3. So both statements are true.

Teacher's Note:
1. Check with 4 and 6: HCF 2, LCM 12, 2 × 12 = 24 = 4 × 6.
2. The LCM equals the product only for co-prime numbers.

Answer: (c) A is true, but R is false.

Explanation:
1. 24 and 25 are consecutive numbers, so their only common factor is 1. A is true.
2. 24 = 2 × 2 × 2 × 3 and 25 = 5 × 5 are composite, not prime. R is false.
3. So the answer is (c).

Teacher's Note:
1. Co-prime numbers need not be prime.
2. Consecutive numbers always have HCF 1.

Answer: (a) (i)-(r), (ii)-(p), (iii)-(q)

Explanation:
1. HCF(32, 48) = 16, so (i)-(r).
2. HCF(21, 28) = 7, so (ii)-(p).
3. HCF(27, 36) = 9, so (iii)-(q).
4. The matching is (i)-(r), (ii)-(p), (iii)-(q).

Teacher's Note:
1. Check by dividing: 32 ÷ 16 = 2, 48 ÷ 16 = 3.
2. After dividing by the HCF, the quotients must be co-prime.

Answer: (a) 12

Explanation:
1. Row A has stars at 0, 4, 8, 12, 16, … (multiples of 4).
2. Row B has stars at 0, 6, 12, 18, … (multiples of 6).
3. The first number after 0 starred in both rows is LCM(4, 6) = 12.

Teacher's Note:
1. Count the ticks between stars to find each step.
2. 24 and 36 are also common, but they are not the first.

Answer: (d) There is a lowest common multiple but no greatest common multiple

Explanation:
1. Common multiples go on for ever, e.g. for 6 and 8: 24, 48, 72, …
2. A never-ending list has no greatest member, but it does have a smallest one, the LCM.
3. So there is a lowest common multiple but no greatest common multiple.

Teacher's Note:
1. Adding the LCM again always gives a bigger common multiple.
2. The LCM is 1 only for the pair 1 and 1.

Answer: (b) 36 kg

Explanation:
1. The bag weight must divide both 108 and 144, so it is a common factor.
2. Fewest bags means the largest such weight, the HCF.
3. 108 = 2 × 2 × 3 × 3 × 3 and 144 = 2 × 2 × 2 × 2 × 3 × 3, so HCF = 2 × 2 × 3 × 3 = 36 kg.

Teacher's Note:
1. 72 does not divide 108, so it cannot work.
2. Check: 108 ÷ 36 = 3 bags and 144 ÷ 36 = 4 bags.

Answer: (b) Only Statement II is true

Explanation:
1. 8 and 9 are co-prime, but neither is prime. Statement I is false.
2. Co-prime numbers have HCF 1, and HCF × LCM = product, so LCM = product. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. One counterexample is enough to show a statement is false.
2. For co-prime numbers, LCM = product.

Answer: (d) A is false, but R is true.

Explanation:
1. Doubling both numbers adds a 2 to both factorisations, so the common part gains a 2 and the HCF doubles.
2. Example: HCF(270, 50) = 10 but HCF(540, 100) = 20. So A is false.
3. R is true. The answer is (d).

Teacher's Note:
1. Try a small example: HCF(6, 8) = 2, HCF(12, 16) = 4.
2. The extra 2 is common to both, so it joins the HCF.

Answer: (c) the product of all the divisors used on the left

Explanation:
1. Each divisor on the left divides both numbers, so it is part of the common factor.
2. The HCF is the product of all the divisors used on the left.
3. Multiplying this by the last pair of quotients as well gives the LCM.

Teacher's Note:
1. Left column multiplied gives the HCF.
2. Left column and the last row multiplied give the LCM.

Answer: (c) 24

Explanation:
1. HCF × LCM = product of the two numbers, so the product = 8 × 96 = 768.
2. Other number = 768 ÷ 32 = 24.
3. Check: HCF(32, 24) = 8 and LCM(32, 24) = 96.

Teacher's Note:
1. Multiply HCF and LCM, then divide by the known number.
2. Always check the answer with both HCF and LCM.

Answer: (d) I and II only

Explanation:
1. Two different primes share no prime, so they are co-prime and LCM = m × n. II is true, III is false.
2. Each prime is at least 2, so m × n is bigger than both m and n. I is true.
3. So the answer is I and II only.

Teacher's Note:
1. Try m = 2, n = 3: LCM = 6.
2. The LCM is never bigger than the product.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. R is true: co-prime means HCF 1, and HCF × LCM = product always holds.
2. Putting HCF = 1 gives 1 × LCM = product, so LCM = product. A is true.
3. R explains A, so the answer is (a).

Teacher's Note:
1. Example: LCM(8, 9) = 72 = 8 × 9.
2. A follows directly from R.

Answer: (b) (i)-(q), (ii)-(r), (iii)-(p)

Explanation:
1. HCF(16, 24) = 8, so (i)-(q).
2. HCF(9, 27) = 9, which is one of the numbers because 9 is a factor of 27, so (ii)-(r).
3. 35 = 5 × 7 and 44 = 2 × 2 × 11 share nothing, so HCF = 1 and (iii)-(p).
4. The matching is (i)-(q), (ii)-(r), (iii)-(p).

Teacher's Note:
1. When one number divides the other, the smaller one is the HCF.
2. No common prime means HCF 1.

Answer: (d) 392

Explanation:
1. The divisors on the left are 2 and 7, and the last pair of quotients is 4 and 7.
2. LCM = product of the divisors and the last quotients = 2 × 7 × 4 × 7 = 392.
3. 14 (= 2 × 7) is the HCF, not the LCM.

Teacher's Note:
1. Check: 392 ÷ 56 = 7 and 392 ÷ 98 = 4.
2. 5488 is the product 56 × 98, not the lowest common multiple.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. The HCF must be a factor of the LCM, but 60 ÷ 8 = 7.5 is not whole. So A is true.
2. R is true, since the LCM is a multiple of the HCF.
3. But 60 is bigger than 8, so R alone does not rule out the pair. R does not explain A, so the answer is (b).

Teacher's Note:
1. The real reason is that the HCF must divide the LCM.
2. A true R must also give the reason to choose (a).