Q) To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B.

1. What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B?
a) 144                    b) 128                     c) 288                          d) 272

2. If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32 , 36) is
a) 2                         b) 4                        c) 6                              d) 8

3. 36 can be expressed as a product of its primes as
a) 2² x 3²                b)2¹ x 3³               c) 2³ x 3¹                     d) 2⁰ x 3⁰

4. 7 x 11 x 13 x 15 + 15 is a
a) Prime number                                  b) Composite number           

c) Neither prime nor composite         d) None of the above

5. If p and q are positive integers such that p = a b² and q= a² b, where a , b are prime numbers, then the LCM (p, q) is
a) a b                       b) a² b²                c) a³ b²                        d) a³ b³

Ans:

STEP BY STEP SOLUTION

1. Minimum number of books:

Here, we need to find the minimum number of books which can be equally distributed among 32 students as well as 36 students.

Therefore we need to find the smallest number divided by 32 and 36.

Hence we will find a number which is multiple of 32 and 36 and is the lowest, hence the LCM.

By Prime factorisation, let’s find factors of 32 and 36:

32 = 2 x 2 x 2 x 2 x 2

36 = 2 x 2 x 3 x 3

Now the LCM will have common factors, uncommon factors of 1st number and uncommon factors of 2nd number.

Hence, the LCM is: (2 x 2) x (2 x 2 x 2) x (3 x 3) = 32 x 9 = 288

Therefore, option c) is correct

2. Find the HCF:

Given that Number 1 x Number 2 = HCF x LCM

∴ HCF =

∴ HCF =

∴ HCF = 4

Therefore, option b) is correct

[Note: you can cross-check: from the factors by Prime factorisation method in part 1, we can see the common factors are 2, 2 and hence HCF is 2 x 2.]

3. Factors of 36:

By Prime factorisation method, we calculated factors of 36 as: 2 x 2 x 3 x 3

Hence, we can write it as: 2²  x 3²

Therefore, option a) is correct

4. Check the expression:

If a natural number has more than two factors, then it is called a composite number.

Here, we can take 15 common and write 7 x 11 x 13 x 15 + 15 as: 15 (7 x 11 x 13 + 1).

Now, we can see that 15 is a factor to the number and 15 itself can be expressed as 3 x 5

So the expression become 5 x 3 x (7 x 11 x 13 + 1)

Since, now the given number has more than two factors, it is a composite number.

Therefore, option b) is correct

Note: Here, we could have solved the expression and we could have written as 3 x 5 x 1002; but since we need to check if it has more than 2 factors or not. That was determined by writing 15 as 3 x 5, hence remaining expression was not solved.

5. Find the LCM:

By Prime factorisation method, factors of p will be: p = a x b x b

Similarly, factors of q will be: q = a x a x b

Now the LCM will have common factors, uncommon factors of 1st number and uncommon factors of 2nd number.

∴ LCM (p, q) = (a x b) x b x a

∴ LCM (p, q) = a² x b²

Therefore, option b) is correct

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