Question

If two positive integers p and q can be expressed as p = 18 a²b⁴ and q = 20 a³b², where a and b are prime numbers, then LCM (p, q) is ______.

विकल्प

• 2 a²b²

• 180 a²b²

• 12 a²b²

• 180 a³b⁴

Answer 1

If two positive integers p and q can be expressed as p = 18 a²b⁴ and q = 20 a³b², where a and b are prime numbers, then LCM (p, q) is 180 a³b⁴.

Explanation:

We are given:

`p = 18 · a^2 · b^2`

`q = 20 · a^3 · b^2`

Step 1: Factor the numerical coefficients

`18 = 2 · 3^2`

`20 = 2^2 · 5`

So we rewrite: 

`p = 2 · 3^2 · a^2 · b^2`

`q = 2^2 · 5 · a^3 · b^2`

Step 2: LCM – take the highest powers of each prime factor

• For 2: max(1, 2) = 2
• For 3: max(2, 0) = 2
• For 5: max(0, 1) = 1
• For a: max(2, 3) = 3
• For b: max(4, 2) = 4

Step 3: Multiply 

LCM = `2^2 · 3^2 · 5 · a^3 · b^4`

LCM = `4 · 9 · 5 · a^3 · b^4 = 180 · a^3 · b^4`

180 a³b⁴