NCERT Exercise 3.5 Solution class six math
In this section question number 1 of NCERT Exercise 3.5 class six math has been solved. In question number 1 there are nine questions have been given. These questions are based on divisibility rule of numbers.
Question (1) Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
Answer False
Explanation
If the sum of the digits of a number is divisible by 9, then the number will be divisible by 9.
Example: (a) 6 is divisible by 3, but it is not divisible by 9.
(b) 51 is divisible by 3, but it is not divisible by 9.
If a number is divisible by 3, then it is not necessary that the number will be divisible by 9 also.
Hence, the given statement, "If a number is divisible by 3, it must be divisible by 9," is false.
(b) If a number is divisible by 9, it must be divisible by 3.
Answer: True
Explanation
Since if a number is divisible by another number then it is divisible by each of the factors of that number.
Since, 3 is one of the factor of 9, thus if a number is divisible by 9 then it is divisible by 3 also.
Thus the given statement, "If a number is divisible by 9, it must be divisible by 3," is true.
(c) A number is divisible by 18, if it is divisible by both 3 and 6
Answer: True
We know that if a number is divisible by another number then it is divisible by each of the factors of that number.
Since, 3 and 6 are factors of 18, thus if a number is divisible by 18 then it will be divisible by both 3 and 6.
Thus the given statement, "A number is divisible by 18, if it is divisible by both 3 and 6," is true.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90
Answer True
Explanation
We know that if a number is divisible by two co-prime numbers then it is divisible by their product also.
Here, 90 is the product of two co-prime numbers 9 and 10, thus if a number is divisible by 9 and 10 both, then it must be divisible by 90 also.
Thus the given statement, "If a number is divisible by 9 and 10 both, then it must be divisible by 90," is true.
(e) If two numbers are co-primes, at least one of them must be prime.
Answer: False
Explanation
Let two numbers 9 and 10
9 and 10 are co-prime and none of them are prime numbers.
Hence, the statement, "If two numbers are co-primes, at least one of them must be prime," is false
(f) All numbers which are divisible by 4 must also be divisible by 8.
Answer: False
Explanation
For a three-digits number if the last two digits forming the number is divisible by 4, then the number will be divisible by 4. On the other hand if the last three digits of a number is divisible by 8 then only the number is divisible by 8.
Example
(a) 84: 84 is divisible by 4 but not divisible by 8.
(b) 156: 156 is divisible by 4 but not divisible by 8.
Thus the given statement, "All numbers which are divisible by 4 must also be divisible by 8," is false.
(g) All numbers which are divisible by 8 must also be divisible by 4
Answer: True
Explanation
We know that if a number is divisible by another number then it is divisible by each of the factors of that number.
Since, 4 is one of the factor of 8, thus the number which is divisible by 8 is divisible by 4.
Thus, the statement, "All numbers which are divisible by 8 must also be divisible by 4," is true.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
Answer: True.
Example (a) Let take a number 45 and 27.
45 and 27 are divisible by 9, and their sum 45 + 27 = 72 is also divisible by 9.
Example (b) 16 and 64 are divisible by 4.
And the sum of 16 and 64 which is equal to 80 is also divisible by 4.
Thus, the given statement, "If a number exactly divides two numbers separately, it must exactly divide their sum," is true.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Answer: True.
Example (a) Let take a number 24 and and 30.
The sum of 24 and 30 = 54
54 is divisible by 3, and 24 and 30 are also separately divisible by 3.
Example (b) The sum of 65 and 15 = 80
80 the sum of 65 and 15 are divisible by 5. Along with this 65 and 15 are separately divisible by 5.
Thus, the given statement, " If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately," is true.