Solve for x.
3x = 6x - 2

The expression of highest degree that exactly divides a given expression is the H.C.F., so the answer to question 1 is (A).
Factoring x 3 − y 3 as ( x − y ) ( x 2 + x y + y 2 ) , the H.C.F. of ( x − y ) 3 and x 3 − y 3 is ( x − y ) , so the answer to question 2 is (A).
Factoring a 2 + a y as a ( a + y ) and a 3 y − a y 3 as a y ( a 2 − y 2 ) , the L.C.M. is a y ( a 2 − y 2 ) , so the answer to question 3 is (D).
If p is a factor of q , the H.C.F. of p and q is p , so the answer to question 4 is (A).
The relation between H.C.F. ( H ), L.C.M. ( K ), and two expressions P and Q is HK = PQ , so the answer to question 5 is (C).

Explanation

Introduction We will solve each question step by step.

Question 1

The expression of highest degree which exactly divides the given expression is the Highest Common Factor (H.C.F.). So the answer is (A).

Question 2

To find the H.C.F. of ( x − y ) 3 and x 3 − y 3 , we first factorize x 3 − y 3 as ( x − y ) ( x 2 + x y + y 2 ) . The H.C.F. of ( x − y ) 3 and ( x − y ) ( x 2 + x y + y 2 ) is ( x − y ) . So the answer is (A).

Question 3

To find the L.C.M. of a 2 + a y and a 3 y − a y 3 , we first factorize a 2 + a y as a ( a + y ) and a 3 y − a y 3 as a y ( a 2 − y 2 ) = a y ( a + y ) ( a − y ) . The L.C.M. of a ( a + y ) and a y ( a + y ) ( a − y ) is a y ( a + y ) ( a − y ) = a y ( a 2 − y 2 ) . So the answer is (D).

Question 4

If p is a factor of q , then q = n p for some integer n . The H.C.F. of p and q is p . So the answer is (A).

Question 5

If H and K are the H.C.F. and L.C.M. of expressions P and Q respectively, then the relation between them is H × K = P × Q . This can be rearranged to HK = PQ , which is equivalent to HK = PQ . Thus, HK = PQ , which means A K = PQ if we assume A is H. So the answer is (C).

Final Answer Therefore, the answers are:

(A) H.C.F.

(A) x − y

(D) a y ( a 2 − y 2 )

(A) p

(C) HK = PQ


Examples
Understanding H.C.F. and L.C.M. is crucial in various real-life scenarios, such as scheduling events or optimizing resource allocation. For instance, if you have two tasks that need to be performed regularly, one every 6 days and another every 8 days, the L.C.M. (24 days) tells you when both tasks will occur on the same day. Similarly, H.C.F. helps in dividing items into equal groups, like distributing 12 pens and 18 pencils equally among students; the H.C.F. (6) indicates the largest number of students who can receive an equal share.

Answered by GinnyAnswer | 2025-07-08