When A and B share no common prime factors, they are co-prime, meaning HCF(A, B) = 1. Substituting HCF(A, B) = 1 into the product formula yields the maximum possible value for the LCM, namely L = A × B.

When A and B share no common prime factors, they are co-prime, meaning HCF(A, B) = 1. Substituting HCF(A, B) = 1 into the product formula yields the maximum possible value for the LCM, namely L = A × B.

If A>B and their LCM is L RULES

Answer: For positive integers $A > B$ with $\text{LCM}(A, B) = L$ : (1) $L$ is a common multiple: $A \mid L$ and $B \mid L$ ; (2) Range: $A \le L \le A \times B$ ; (3) If $B$ divides $A$ , then $L = A$ ; (4) General relation: $\text{HCF}(A, B) \times L = A \times B$ ; (5) If $\text{HCF}(A, B) = 1$ , then $L = A \times B$ .

Step-by-step solution

Step 1: Define divisibility and bounding rules

By definition, the least common multiple $L = \text{LCM}(A, B)$ of two positive integers is a multiple of both $A$ and $B$ . Since $A$ and $B$ are positive non-zero integers with $A > B$ , the multiple $L$ must be at least as large as the greater number, which gives $L \ge A > B$ .

Step 2: State the special case of divisibility

If the smaller number $B$ divides the larger number $A$ completely, then $A$ is already a common multiple of both $A$ and $B$ . Because $A$ is the smallest possible non-zero multiple of $A$ , the least common multiple is exactly $L = A$ .

Step 3: Relate LCM and HCF to the product of numbers

For any two positive integers $A$ and $B$ , the product of their highest common factor and their least common multiple equals the product of the two numbers. This fundamental identity allows us to compute $L$ or bound it using their greatest common divisor.

Step 4: State the rule for co-prime numbers

When $A$ and $B$ share no common prime factors, they are co-prime, meaning $\text{HCF}(A, B) = 1$ . Substituting $\text{HCF}(A, B) = 1$ into the product formula yields the maximum possible value for the LCM, namely $L = A \times B$ .

Solve your own maths question free →