Question

For positive integers \[x\], let the symbol $\underline{\underline x} $​​ represent the sum of the digits of \[x\].
For example, $\underline{\underline {74}} = 7 + 4 = 11$. If $n$ is a positive integer and $\underline{\underline n} = \underline{\underline {33}} + \underline{\underline {17}} $. which of the following could be the value of $n$?
$A)14$
$B)34$
$C)51$
$D)65$
$E)86$

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
All integer numbers are basically of three types; zero, positive numbers, negative numbers.
Positive numbers are these numbers that are prefixed with a plus sign (+).Sum is the adding of two or more integers or numbers.

Complete step-by-step solution:
Given that the sum of the digits of $n$ is equal to the sum of the digits of $33 + $ sum of the digits of $17$
Since the symbol $\underline{\underline x} $​​ represents the sum of the digits of \[x\]. And \[x\] is a positive integer.
Addition Rule is if we add two integers with the same sign then the absolute value and the result is assigned the same sign as the both the values. For example, $7 + 4 = 11$
Since from the given question we can able to write $11$ as $7 + 4$ and then make it as $74$ (sum of the digits from left side is placed first and right-side number is placed second)
Hence by this we can able to reframe $\underline{\underline n} = \underline{\underline {33}} + \underline{\underline {17}} $as $\underline{\underline n} = 3 + 3 + \underline{\underline {17}} $($\underline{\underline {33}} $ can be rewrite as 3 plus 3)
Similarly, $\underline{\underline {17}} $ can be reframed as 1 plus 7. Thus $\underline{\underline n} = 3 + 3 + 1 + 7$
Therefore $\underline{\underline n} = 14$ is the sum of the digits.
First in option A $n = 14$that means $\underline{\underline n} = 1 + 4 = 5$ {since the sum of the n is $14$ it is wrong}
For option B $n = 34$ that means $\underline{\underline n} = 3 + 4$ the sum of n is 7 it is wrong
For option C $n = 51$ then sum of the digits is $6$ and for option D $n = 65$ sum of the digits is $11$
Hence in option E $n = 86$ and $\underline{\underline n} = 8 + 6$ which gives the sum of digits as $14$
Therefore option E $n = 86$ which gives the sum of number $\underline{\underline n} = 14$

Note: The sum of the digits only calculated if it is given as $\underline{\underline n} $.We can also able first simplify the given answer by sum of the digits formula like $14$ as $1 + 4$ and then we find the given question which yields the same desired results as above.