Are all numbers good?

Number Theory Level 2

A two-digit positive integer, whose sum of digits and product of digits add up to the number itself is called a good number . Find the number of good numbers .

Details and assumptions:

I am asking for only the first sum of digits, not the digital sum, which is an iterated process.
Both digits need not be distinct, but a leading zero is not allowed.

$\boxed{\text{Difficulty: }\color{#D61F06}{\text{Small, but pointy.}} \quad \Large \color{#D61F06}{\dagger~ \dagger~ }\color{grey}{\dagger\dagger \dagger}}$

The answer is 9.

2 solutions

Mahdi Raza
Aug 19, 2020

Use Algebra

$\begin{aligned} a + b + ab &= 10a + b \\ a(1 + b) &= 10a \\ 1 + b &= 10 & \blue{\text{for } a \ne 0} \\ b&= 9 & \blue{\text{for } a \ne 0} \end{aligned}$

All values of $a$ can take place except 0, which are $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ , hence there are $\boxed{9}$ solutions

Python

total = 0

for a in range(1,10):
    for b in range(0,10):
        if a+b + a*b == 10*a + b:
            total = total + 1

print(total)

I don't understand the problem. Good number $=\overline{ab}+a+b+ab$

A Former Brilliant Member - 9 months, 3 weeks ago

A Former Brilliant Member
Aug 20, 2020

Let the first digit be $x$ and the second be $y$ , so

$x+y+xy=10x+y \rightarrow y=9$

The second digit must be $9$ for a two digit positive integer to be a good number. The first digit is such that $0<x<10$ , so there are $9$ possible first digits and hence $9$ possible two digit good numbers.

Are all numbers good?

The answer is 9.

2 solutions

Use Algebra

Python

0 pending reports