Classify each of the following statements as true or false where a and b are dis
ID: 3113374 • Letter: C
Question
Classify each of the following statements as true or false where a and b are distinct natural numbers. a. LCM(a,b)IGCD(ab) b. LCM(a b)lab c. GCD(a,b)sa d. LCM(a,b)2 a a. Choose the correct answer below O A. True. The LCM of two numbers is always a divisor of the GCD of those numbers O B. False. The LCM of two numbers is only a divisor of the GCD of those numbers if they have no common factors o c. False. The LCM of two numbers is only a divisor of the GCD of those numbers if they have at least one common factor D. False. The LCM of two numbers is never a divisor of the GCD of those numbers, however, the GCD is a divsor of the LCM b. Choose the correct answer below O A. False. The LCM of two numbers is never a divisor of the product of those numbers, however, the product is a divisor of the LCM 0 B. True. The product of two numbers will always have the same factors as the LCM O C. False. When the product of two numbers is equal to the LCM, the LCM is not a divisor of the productExplanation / Answer
In the first question:
a.LCM(a,b)=GCD(a,b) is false because LCM(4,8) is 8 but GCD(4,8) is 4
b. LCM(a,b)=ab is false. eg. LCM(4,8) is not 4*8=32, it is 8
c. GCD(a,b)=< a is true. for eg. GCD(3,9) is 3 and it is less than or equal to 3. This will always be the case because GCD will always be either equal to or less than the numbers.
d. LCM(a,b)=> a is true. For eg. LCM(3,9) is 9 and here 3 is not smaller than 9.
C. A. Correct because as GCD(3,9) is 3 and it is equal to 3 . If we take a > b, GCD( 9,3) is 3 is still less than a i.e.9
B. Correct because GCD(3,9) is 3 and it is equal to 3 i.e. a but not less a i.e. 3 is not greater than b i.e.9
C. InCorrect because GCD will always be less than either or both and equal to either or both of them. Eg. GCD(8,24) is 8. It is less than 24 but equal to 8. Eg. GCD(64 ,48) is 16. Here 16 is less than both of them.
D. We can not say whether the statement is corret or incorrect. It is a null statement because GCD(64,48) is 32 is neither greater than nor equal to 64 or 48. However GCD(3,9) is 3 which is not greater than 3 i.e.a but equal to 3 i.e a. Therefore it is a null statement.
d. Let us take three examples, 1. LCM (a,b)= LCM(4,8)=8
2. LCM(a,b)=LCM(8,4)=8
3. LCM(a,b)=LCM( 2,5)=10
4. LCM(a,b)=LCM(2,2)=2
A. Taking eg.2, the statement is true but stated false here, so it is incorrect.
B. Taking eg.1, the statement is false and stated false here as well, so it is correct.
C. Takingneg. 1,3 and 4, the statement is true but stated false here, so it is incorrect.
D. Taking all the baove examples, this statement is false because the true statement is greater than or equal to either or both of them. It has been stated true here, so it is incorrect statement.