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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-primes, at least 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.
To do:
We have to find which of the given statements is true.
Solution :
(a) We know that,
If a number is divisible by another number, it is divisible by its factors.
Therefore,
If a number is divisible by 9 it is divisible by 3 but the converse is not necessarily true.
For example,
12 and 15 are divisible by 3 but not by 9.
Therefore, if a number is divisible by 3 it may not be divisible by 9.
The given statement is false.
(b) We know that,
If a number is divisible by another number, then it is divisible by its factors.
Therefore,
If a number is divisible by 9 it is divisible by 3
The given statement is true.
(c) A number is divisible by 18 if it is divisible by both 9 and 2.
The given statement is false.
(d) A number is divisible by 90 if it is divisible by both 9 and 10.
The given statement is true.
(e) When two numbers have no common factors other than 1, then the numbers are co-prime.
8 and 9 are co-primes but both 8 and 9 are not prime numbers.
(f) For example,
12 is divisible by 4 but is not divisible by 8.
Therefore, the given statement is false.
(g) 4 is a factor of 8.
This implies,
All numbers which are divisible by 8 must also be divisible by 4.
Therefore, the given statement is true.
(h) For example,
2 divides 6 and 10.
$6 + 10 = 16$
$16$ is also divisible by 2.
Therefore, the given statement is true.
(i) $10+30=40$ is divisible by 4 but both 10 and 30 are not divisible by 4.
The given statement is false.