Question

The digits of a three-figure number are consecutive odd numbers. The number is 51 less than thirty times the sum of its digits. What is the number?
A.975
B.579
C.759
D.597

Answer 1

Hint: We can take the 3 digits as $x - 2,x$ and $x + 2$. Then the 3-figure number will be $\left( {x - 2} \right) \times 100 + x \times 10 + \left( {x + 2} \right) \times 1$. Then we can take the sum of the digits and form an equation as per the relation given in the question. We get the value of x by solving the equation and find the digits to find the number.

Complete step-by-step answer:
We are given that digits of a 3-figure number are consecutive odd numbers. So, we can take the digits to be $x - 2,x$ and $x + 2$.
We can take the sum of the digits by adding the digits. So, we get,
$
   \Rightarrow Sum = \;x - 2 + x + x + 2 \\
   \Rightarrow Sum = 3x \\
 $
As the digits are $x - 2,x$ and $x + 2$, we can write the digit as the sum of the products of the digit with the value of its place.
Then the required number is given by $\left( {x - 2} \right) \times 100 + x \times 10 + \left( {x + 2} \right) \times 1$… (1)
On simplification, we get the number as
$
   \Rightarrow 100x - 200 + 10x + x + 2 \\
   \Rightarrow 111x - 198 \\
 $
From the question, we know that the number is 51 less than 30 times the sum of digits. So, we can write the above statement in the form of a mathematical equation.
30 times sum – 51 = the number
We found that the sum is 3x and the number is $111x - 198$.
So we get,
$ \Rightarrow 30 \times 3x - 51 = 111x - 198$
On simplification we get,
$ \Rightarrow 90x - 51 = 111x - 198$
On taking like terms on the same side we get,
$ \Rightarrow 111x - 90x = 198 - 51$
On simplification we get,
$ \Rightarrow 21x = 147$
On dividing the equation by 21 we get,
$ \Rightarrow x = \dfrac{{147}}{{21}}$
On simplification we get,
$ \Rightarrow x = 7$
As x=7, we can write the other digits by substituting the value of x in
$x - 2,x$ and $x + 2$
 $
   \Rightarrow 7 - 2,7,7 + 2 \\
   \Rightarrow 5,7,9 \\
 $
Therefore, the digits are 5, 7 and 9.
We can find the required number by substituting for x=7 in equation (1).
\[
   \Rightarrow \left( {7 - 2} \right) \times 100 + 7 \times 10 + \left( {7 + 2} \right) \times 1 \\
   \Rightarrow 5 \times 100 + 7 \times 10 + 9 \times 1 \\
   \Rightarrow 500 + 70 + 9 \\
   \Rightarrow 579 \\
 \]
Therefore, the required number is 579.
So, the correct answer is option B.

Note: This problem deals with basic number theory and mathematical modeling. Mathematical modeling is the method of making mathematical equations from statements. We took the digits as $x - 2,x$ and $x + 2$ because the odd numbers come in a difference of 2 and taking the numbers like this makes it easier for us to take the sum and do other calculations. We assume that the order of digits in the number is in the increasing order. We can check whether our assumption is right or wrong after getting the value of x and other digits. In this case we got the digits as perfect odd numbers, so our assumption was true. If we got an even number or any solution that is not an integer, we must consider changing the order of the digits in the number. We can verify our answer by checking whether the number satisfies the condition given in the question.