question_answer 31)

Which of the following statements are true? (a) If a number is divisible by 3, it must be divisible by 9. (b) If a number is divisible by 9, it must be divisible by 3. (c) A number is divisible by 18, if it is divisible by both 3 and 6. (d) If a number is divisible by 9 and 10 both, then it must be divisible by 90. (e) If two numbers are co-prime, atleast one of them must be prime. (f) All numbers which are divisible by 4 must also be divisible by 8. (g) All numbers which are divisible by 8 must also be divisible by 4. (h) If a number exactly divides two numbers separately, it must exactly divide their sum. (i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.                

Answer:

                (a) False, because there are plenty of numbers, which are divisible by 3 but not divisible by 9. e.g. 30 is divisible by 3, but not divisible by 9. (b) True, because if a number is divisible by any number, then it is divisible by each factor of that number. Here, 3 is a factor of 9. e.g. \[\frac{27}{9}=3\]and\[\frac{27}{3}=9\] (c) False, e.g. Number 30 is divisible by 3 and 6 both but not divisible by 18. (d) True, because if a number is divisible by two co-prime numbers, then it is divisible by their product also. (e) False, we know that, two numbers having only 1 as a common factor are called co-prime numbers. So, it is not necessary that one of them must be prime. e.g. numbers 8 and 15 are co-prime numbers, since both have only 1 as a common factor, but no one is a prime number. (f) False, e.g. number 36 is divisible by 4 but not divisible by 8. (g) True because if a number is divisible by any number, then it is divisible by each factor of that number. Here, 4 is a factor of 8. So, all numbers divisible by 8 must also be divisible by 4. e.g. Number 56 is divisible by 8 as well as divisible by 4. (h) True, if two given numbers are divisible by a number, then their sum is also dividible by that number, e.g. number 13 is exactly divides number 52 and 65 also divide their sum 117. False, e.g. Number 5 is exactly divides the sum of number 2 and 3 but not exactly divides these two numbers.