Decoding the Number Coding Method

The question describes a method where a number is represented by three digits. These digits are the remainders when the original number is divided by 3, 7, and 11, respectively.

For example, the number 53 results in the code (2, 4, 9) because:

• $53 \div 3 = 17 \text{ remainder } 2$
• $53 \div 7 = 7 \text{ remainder } 4$
• $53 \div 11 = 4 \text{ remainder } 9$

We need to find the least number that corresponds to the code (2, 2, 2).

Finding the Least Number for Code (2, 2, 2)

A number with the code (2, 2, 2) must satisfy the following conditions simultaneously:

• The number leaves a remainder of 2 when divided by 3. Mathematically, $N \equiv 2 \pmod{3}$.
• The number leaves a remainder of 2 when divided by 7. Mathematically, $N \equiv 2 \pmod{7}$.
• The number leaves a remainder of 2 when divided by 11. Mathematically, $N \equiv 2 \pmod{11}$.

Since the remainder (2) is the same for all divisors (3, 7, 11), the number $N$ must be 2 more than a multiple of the least common multiple (LCM) of 3, 7, and 11.

The numbers 3, 7, and 11 are prime numbers. Therefore, their LCM is simply their product:

$\text{LCM}(3, 7, 11) = 3 \times 7 \times 11 = 21 \times 11 = 231$.

So, the number $N$ must satisfy the condition:

$N \equiv 2 \pmod{231}$

This means $N$ can be expressed in the form $N = 231k + 2$, where $k$ is a non-negative integer.

To find the *least* such number, we start with the smallest possible value for $k$, which is $k=0$.

• If $k=0$, $N = 231 \times 0 + 2 = 2$.
• If $k=1$, $N = 231 \times 1 + 2 = 233$.
• If $k=2$, $N = 231 \times 2 + 2 = 462 + 2 = 464$.

The possible numbers are 2, 233, 464, and so on. We are looking for the least number among the given options that fits this pattern. The smallest number generated this way, which is also a three-digit number typically expected in such problems, is 233.

Verifying the Least Number

Let's check if the number 233 yields the code (2, 2, 2):

• $233 \div 3 = 77 \text{ remainder } 2$
• $233 \div 7 = 33 \text{ remainder } 2$
• $233 \div 11 = 21 \text{ remainder } 2$

The remainders are indeed (2, 2, 2). Since 233 is the smallest number greater than 2 that fits the condition $N \equiv 2 \pmod{231}$, it is the least number with the specified code.