Question

LCM(P, 375, 225) = $3^3$ x $5^3$

HCF(P, 375, 225) = 3 x 5

Which of the following could be the value of P?

• A. 15
• B. 135
• C. 675
• D. P cannot be uniquely determined

Answer 1

Answer: (B)

135

Answer 2

Let

$P = 3^a \cdot 5^b, 375 = 3^1 \cdot 5^3, 225 = 3^2 \cdot 5^2$.

LCM Condition:

$\text{LCM}(P,375,225)=3^{\max(a,1,2)}\cdot 5^{\max(b,3,2)}=3^3\cdot5^3$.

• For prime 3: $\max(a,1,2)=3 \implies a=3$.
• For prime 5: $\max(b,3,2)=3 \implies b\leq3$ (but see HCF condition next).

HCF Condition:

$\text{HCF}(P,375,225)=3^{\min(a,1,2)}\cdot 5^{\min(b,3,2)}=3^1\cdot5^1$.

• For prime 3: $\min(3,1,2)=1$ (which is satisfied).
• For prime 5: $\min(b,3,2)=1 \implies b=1$ (since if $b>1$, the minimum becomes higher than 1).

Thus,

$P = 3^3 \cdot 5^1 = 27 \cdot 5 = 135$.

Checking the options, only option 2 (135) meets the conditions.