The digits of a three-figure number are consecutive odd number. The number is 51 less than thirty times the sum of its digits. What is the number?

1. Break Down the Problem

Let's denote the three-digit number as $xyz$, where $x$, $y$, and $z$ are the digits. Since the digits are consecutive odd numbers, we can express them as:

• Let the smallest digit $x = 2n + 1$
• Then, $y = 2n + 3$
• And $z = 2n + 5$

2. Relevant Concepts

The number formed by the digits $xyz$ can be expressed as: $N = 100x + 10y + z$ The sum of the digits is: $S = x + y + z$ According to the problem, the relationship between the number $N$ and the sum of its digits $S$ is given by the equation: $N = 30S - 51$

3. Analysis and Detail

First, find the digits in terms of $n$:

• The sum of the digits: $S = (2n + 1) + (2n + 3) + (2n + 5) = 6n + 9$

• The number $N$: Substituting $x$, $y$, and $z$: $N = 100(2n + 1) + 10(2n + 3) + (2n + 5) = 200n + 100 + 20n + 30 + 2n + 5 = 222n + 135$

Setting up the equation

Now we substitute $N$ and $S$ in the main equation: $222n + 135 = 30(6n + 9) - 51$

Solving the equation

1. Expand the right side: $30(6n + 9) = 180n + 270$ So, $222n + 135 = 180n + 270 - 51$ $222n + 135 = 180n + 219$

2. Rearranging gives: $222n - 180n = 219 - 135$ $42n = 84$ $n = 2$

4. Verify and Summarize

Now substitute $n = 2$ back into the expressions for $x$, $y$, and $z$:

• $x = 2(2) + 1 = 5$
• $y = 2(2) + 3 = 7$
• $z = 2(2) + 5 = 9$

Thus, the number $N$ is: $N = 579$

Final Answer

The three-figure number is $579$.