Let P= QQQ be a 3-digit number. What is the HCF of P and 481?

Correct Answer: Option 37

To find the Highest Common Factor (HCF) of the 3-digit number $P$ and $481$, let us first analyze the structure of $P$ and find the prime factorization of $481$.

Step 1: Express the 3-digit number P in algebraic form.

The number $P$ is given as a 3-digit number with identical digits, $P=QQQ$, where $Q$ is a single non-zero digit ($Q\in \{1,2,3,4,5,6,7,8,9\}$).

We can write $P$ as:

$P=100Q+10Q+Q=111\times Q$

Now, let us find the prime factorization of $111$:

$111=3\times 37$

Thus, $P$ can be written as:

$P=3\times 37\times Q$

Step 2: Find the prime factorization of 481.

We check for divisibility of $481$ by prime numbers:

Testing division by $13$:

$481÷13=37$

Since both $13$ and $37$ are prime numbers, the prime factorization of $481$ is:

$481=13\times 37$

Step 3: Calculate the HCF of P and 481.

We want to find $\text{HCF}(P,481)=\text{HCF}(3\times 37\times Q,13\times 37)$.

Notice the prime factors of each term:

• Prime factors of $481$ are strictly $13$ and $37$.

• The factor $37$ is common to both $P$ and $481$.

• The factor $13$ is not present in $111$ ($3\times 37$). Also, since $Q$ is a single-digit integer from $1$ to $9$, $Q$ cannot be divisible by $13$.

Therefore, the highest common factor shared between $P$ and $481$ is strictly $37$.