A number is three times the sum of its digit. What is the number?

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A number is three times the sum of its digit. What is the number?👇🏾👇 🏾

👇🏾👇 🏾

Let the number be denoted as

$𝑎$ $𝑏$ $‾$ ab

, where

$𝑎$ $a$ and $𝑏$ $b$ are the digits. Since $𝑎$ $𝑏$ $‾$ ab

represents a two-digit number, it can be expressed mathematically as:

$10$ $𝑎$ $+$ $𝑏$ $10 a + b$ $𝑎$ $+$ $𝑏$ $a + b$ $10$ $𝑎$ $+$ $𝑏$ $=$ $3$ $($ $𝑎$ $+$ $𝑏$ $)$ $10 a + b = 3 (a + b)$ $10$ $𝑎$ $+$ $𝑏$ $=$ $3$ $𝑎$ $+$ $3$ $𝑏$ $10 a + b = 3 a + 3 b$ $3$ $𝑎$ $+$ $3$ $𝑏$ $3 a + 3 b$ from both sides: $10$ $𝑎$ $+$ $𝑏$ $-$ $3$ $𝑎$ $-$ $3$ $𝑏$ $=$ $0$ $10 a + b - 3 a - 3 b = 0$ $7$ $𝑎$ $-$ $2$ $𝑏$ $=$ $0$ $7 a - 2 b = 0$ $𝑏$ $b$ : $7$ $𝑎$ $=$ $2$ $𝑏$ $⟹$ $𝑏$ $=$ $7$ $𝑎$ $2$ $7 a = 2 b ⟹ b =$ 27a

According to the problem, the number is three times the sum of its digits. The sum of its digits is:

Thus, we have the equation:

Expanding and simplifying this equation:

Subtract

This simplifies to:

Solving for

Since

$𝑎$ $a$ and $𝑏$ $b$ are digits, $𝑎$ $a$ must be chosen such that $𝑏$ $b$ is an integer between 0 and 9. Therefore, $7$ $𝑎$ $7 a$ must be divisible by 2, meaning $𝑎$ $a$ must be even. $𝑎$ $a$ : $𝑎$ $=$ $2$ $𝑏$ $=$ $7$ $\times$ $2$ $2$ $=$ $7$ $⟹$ $Number is$ $27$ $b =$ 27×2

We test even values of

$a = 2$ :
$𝑎$
$a = 4$ :
$𝑎$

No need to test higher even values of

So, we verify the candidate number 27:

A number is three times the sum of its digit. What is the number?

A number is three times the sum of its digit. What is the number?👇🏾👇 🏾

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