Question

EXAMPLE 3 For any natural number a, we define aN = {ax : x € N). If b, c, d €N such that
bN U cN = dN, then prove that d is the l.c.m. of b and c.​

Answer 1

Given: Natural number a, aN = {ax : x € N)

To find:  Prove that d is the l.c.m. of b and c.​

Solution:

• Now we have given that b, c and d belongs to N such that bN ∪ cN = dN

• Here in order to prove that d is the lcm of b and c , we need to understand that the given condition should be bN ∩ cN

• So now, let y belongs to N:

            aN = { ay }

            bN = { by }

            cN = { cy }

            dN = { dy }

• Also we have given that bN ∩ cN = dN , so:

• Considering any element, say z, containing in dN is an element which is present in the other two sets, that is bN and cN .

• So we can say that z is a multiple of b as well as c also.  So here we can also conclude that z should also be the lcm of b and c.

• But we know that z also belongs to dN(as mentioned above), so hence it is a multiple of d also.

Answer:

              So therefore d is lcm of b and c.