Question

# If $n$ is the smallest number such that $n+2n+3n+...+99n$ is a perfect square, then the number of digits in ${{n}^{2}}$ isa.$1$b.$2$c.$3$d.None of these

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Hint: To find the minimum value of $n$ such that the value of sum $n+2n+3n+...+99n$ is a perfect square, use the formula for the sum of $k$ consecutive positive integers as $\sum\limits_{i=1}^{k}{i}=\dfrac{k\left( k+1 \right)}{2}$ to find the sum of $n+2n+3n+...+99n$. Find the terms needed to be multiplied to make the given value of sum a perfect square. Square the calculated value of $n$ and count the digits in the value of ${{n}^{2}}$.

We have to find the smallest value of integer $n$ such that the value of $n+2n+3n+...+99n$ is a perfect square. Further, we have to calculate the digits in the number ${{n}^{2}}$.
We can rewrite $n+2n+3n+...+99n$ as $n\left( 1+2+3+...+99 \right)$.
We have to find the value of $1+2+3+...+99$.
We know that the formula for sum of $k$ consecutive positive integers is $\sum\limits_{i=1}^{k}{i}=\dfrac{k\left( k+1 \right)}{2}$.
Substituting $k=99$, we have $1+2+3+...+99=\dfrac{99\times 100}{2}=99\times 50$.
Thus, we have $n+2n+3n+...+99n=n\left( 99\times 50 \right)$.
We observe that $n\left( 99\times 50 \right)$ is not a perfect square. We have to make it a perfect square. Factorizing the term $n\left( 99\times 50 \right)$, we have $n\left( 99\times 50 \right)=n\left( 9\times 11\times 2\times 25 \right)$.
We observe that $9\times 25$ is already a perfect square. Thus, the minimum value of $n$ should be $11\times 2$ to make $n\left( 9\times 11\times 2\times 25 \right)$ a perfect square.
Thus, we have the value of $n$ as $n=11\times 2=22$.
So, the value of ${{n}^{2}}$ will be ${{n}^{2}}=484$.
Hence, the number of digits in ${{n}^{2}}$ is $3$, which is option (c).

Note: It’s necessary to use the formula for calculating the sum of $k$ consecutive positive integers. Also, it’s necessary to keep in mind that the value of $n$ has to be minimum to get a perfect square, otherwise, we will get an incorrect answer. A perfect square is a number obtained by multiplying a whole number by itself. The perfect square numbers must end with digits $1,4,5,6,9$. Perfect squares never end with digits $2,3,7,8$.

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