Exploring Digit Properties: Solving for Two-Digit Numbers

Mathematics often presents intriguing problems that involve the properties of digits within numbers. In this blog post, we'll explore a problem where we need to find all two-digit numbers \( N \) that satisfy a specific equation involving the product and sum of their digits. Let's dive into the details and solve this problem step-by-step.

Let ` P(n) ` and ` S(n) ` denote the product and sum, respectively, of the digits of a positive integer ` n `. Determine all two-digit numbers ` N ` that satisfy the equation:

` P(N) + 2S(N) = N ` 

For example, `P(23)=2\times 6` and `S(23)=2+3=5`. 

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Answer: `14, 36, 77`

Explanation: 

•  Let ` N = 10a + b `, where ` a ` and ` b ` are the tens and units digits of ` N `, respectively.
  Here, ` a ` and ` b ` are integers such that ` 1 \leq a \leq 9 ` and ` 0 \leq b \leq 9 `.
• Then ,
• The product of the digits ` P(N) ` is
  
  `P(N) = a \cdot b`
• The sum of the digits ` S(N) ` is:
  
       `S(N) = a + b `
• Then, the equation ` P(N) + 2S(N) = N ` becomes

     `a \cdot b + 2(a + b) = 10a + b`

     `     a \cdot b + 2a + 2b = 10a + b     `

     `     a \cdot b + 2a + 2b - b = 10a     `

     `     a \cdot b + 2b - b = 10a - 2a     `

     `a \cdot b + b = 8a     `

     `b(a + 1) = 8a   `

`     b = \frac{8a}{a + 1}`

• For ` b ` to be an integer, ` \frac{8a}{a + 1} ` must be an integer.
• Check values of ` a ` from 1 to 9 to find valid solutions:
• ` a = 1 `: ` b = \frac{8 \cdot 1}{1 + 1} = 4 `
• ` a = 2 `: ` b = \frac{8 \cdot 2}{2 + 1} = \frac{16}{3} ` (not an integer)
• ` a = 3 `: ` b = \frac{8 \cdot 3}{3 + 1} = 6 `
• ` a = 4 `: ` b = \frac{8 \cdot 4}{4 + 1} = \frac{32}{5} ` (not an integer)
• ` a = 5 `: ` b = \frac{8 \cdot 5}{5 + 1} = \frac{40}{6} ` (not an integer)
• ` a = 6 `: ` b = \frac{8 \cdot 6}{6 + 1} = \frac{48}{7} ` (not an integer)
• ` a = 7 `: ` b = \frac{8 \cdot 7}{7 + 1} = 7 `
• ` a = 8 `: ` b = \frac{8 \cdot 8}{8 + 1} = \frac{64}{9} ` (not an integer)
• ` a = 9 `: ` b = \frac{8 \cdot 9}{9 + 1} = 7.2 ` (not an integer)
•    The valid pairs `(a, b)` are `(1, 4)`, `(3, 6)`, and `(7, 7)`.
•    Therefore, the valid two-digit numbers ` N ` are

`N = 10a + b \implies 14, 36, 77`