Question 1
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) If a number is divisible by 18, it must be 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.
Question 1
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) If a number is divisible by 18, it must be 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.
(a) 6 is divisible by 3 but not by 9. Hence, the statement is false.
(b) Since 9 itself is divisible by 3, so any number which is divisible by 9 will be divisible by 3. Hence, the statement is true.
(c) Since, 18 = 3 x 6. So for any number to be divisible by 18, it must be divisible by 3 and 6. Hence, the statement is true.
(d) Since, 9 x 10 = 90. So, if a number is divisible by 9 and 10 both, then it will be divisible by 90. Hence, the statement is true.
(e) As 9 and 4 are co-prime but both 9 and 4 are composite numbers. Hence, the statement is false.
(f) As 12 is divisible by 4 but not by 8, so any number divisible by 4 need not be divisible by 8. Hence, the statement is false.
(g) As 8 = 4 x 2, so any number divisible by 8 have to be divisible by 4. Hence, the statement is true.
(h) Since the two numbers are divisible by any number means that these two numbers are the multiples of the third numbers. If two numbers are multiple of any number then their sum is also a multiple of that number and hence will be divisible by that number. Hence, the statement is true.
(i) This statement is not necessarily true as 9 is divisible by 3 but 9 = 4 + 5 and 4, 5 are not divisible by 3.
Statements (b), (c), (d), (g) and (h) are true.
(a) 6 is divisible by 3 but not by 9. Hence, the statement is false.
(b) Since 9 itself is divisible by 3, so any number which is divisible by 9 will be divisible by 3. Hence, the statement is true.
(c) Since, 18 = 3 x 6. So for any number to be divisible by 18, it must be divisible by 3 and 6. Hence, the statement is true.
(d) Since, 9 x 10 = 90. So, if a number is divisible by 9 and 10 both, then it will be divisible by 90. Hence, the statement is true.
(e) As 9 and 4 are co-prime but both 9 and 4 are composite numbers. Hence, the statement is false.
(f) As 12 is divisible by 4 but not by 8, so any number divisible by 4 need not be divisible by 8. Hence, the statement is false.
(g) As 8 = 4 x 2, so any number divisible by 8 have to be divisible by 4. Hence, the statement is true.
(h) Since the two numbers are divisible by any number means that these two numbers are the multiples of the third numbers. If two numbers are multiple of any number then their sum is also a multiple of that number and hence will be divisible by that number. Hence, the statement is true.
(i) This statement is not necessarily true as 9 is divisible by 3 but 9 = 4 + 5 and 4, 5 are not divisible by 3.
Statements (b), (c), (d), (g) and (h) are true.
