Spy Number

What Are Spy Numbers?

A Spy Number is a special type of number in which the sum of its digits is equal to the product of its digits. These numbers are interesting from a mathematical perspective and often used as examples to understand the relationship between digits.

Mathematical Representation

Let the number be represented as \( N = d_1d_2d_3\ldots d_k \), where \( d_1, d_2, d_3, \ldots, d_k \) are the digits of \( N \). The sum of its digits is:

The product of its digits is:

If:

Then \( N \) is a Spy Number.

Example of a Non-Spy Number

Consider the number \( 123 \):

• Digits: \( 1, 2, 3 \)
• Sum: \( 1 + 2 + 3 = 6 \)
• Product: \( 1 \times 2 \times 3 = 6 \)

Since the sum and product are equal, \( 123 \) is a Spy Number.

Example of a Spy Number

Consider the number \( 112 \):

• Digits: \( 1, 1, 2 \)
• Sum: \( 1 + 1 + 2 = 4 \)
• Product: \( 1 \times 1 \times 2 = 2 \)

Since the sum and product are not equal, \( 112 \) is not a Spy Number.

Algorithm to Check Spy Numbers

Here is a simple algorithm to check if a number is a Spy Number:

1. Extract the digits of the number.
2. Calculate the sum of its digits.
3. Calculate the product of its digits.
4. Compare the sum and product.
   - If equal, it is a Spy Number.
   - Otherwise, it is not.
    

Conclusion

Spy Numbers demonstrate an interesting property of digits. They can be used to illustrate mathematical patterns and make learning fun for students. Try finding other Spy Numbers yourself!