Real Numbers 1

1. A number q is prime factorised as 3² × 7² × b, where b is a prime number other than 3 and 7. Based on the above information, two statements are given below – one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).

Assertion (A): q is definitely an odd number.
Reason (R): 3² × 7² is an odd number.

2. If a =  2³ × 3 , b = 2×3 × 5, c = \( 3^n \) × 5 and LCM(a,b,c) =2³ ×3²× 5, then  n = —–

3. The HCF of k and 93 is 31, where k is a natural number. Which of these can be true for some values of k ?

(i) K is a multiple of 31.                     (ii) k is a multiple of 93
(ii) K is an even number.                  (iv) k is an odd number

4. Let p be a prime number and k be a positive integer. If p divides k², then which of these is definitely divisible by p ?

(i) k/2                         (ii) k                            (iii) 7k                      (iv) k³

5. Two statements are given below – one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).
12
Statement A (Assertion): If product of two numbers is 12960 and their HCF is 12, then their LCM is 1080.
Statement R(Reason): HCF is always a factor of LCM

6. In a formula racing competition , the time taken by two racing cars A and B to compete one round of the track is 30 minutes and p minutes respectively. If the cars meet again at the starting point for the first time after 90 minutes and HCF (30,p)=15, then the value of p is

7. If two positive integers a and b are written as a = x³y² and = xy³, where x and y are prime numbers , then the result obtained by dividing the product of the positive integers by the LCM(a,b) is

8. The LCM of smallest two-digit composite number and smallest composite number is:

9. If sum of two numbers is 1215 and their HCF is 81, then the possible number of pairs of such numbers are

10. The ratio between the LCM and HCF of 5, 15, 20 is: