Consider the following sum : @37 + 46@ + 5@3 = 1@@@@ In the above sum,

Consider the following sum :
@37 + 46@ + 5@3 = 1@@@@
In the above sum, @ stands for which one among the following?

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For all digits

For even digits only

For odd digits only

For no digits

Answer is Right!

Answer is Wrong!

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UPSC CISF-AC-EXE – 2024

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In the given sum, ‘@’ stands for any digit from 0 to 9, which means for all digits.

Let the sum be represented by the equation:
$(100 \times @ + 30 + 7) + (100 \times 4 + 60 + @) + (100 \times 5 + 10 \times @ + 3) = (1000 \times 1 + 100 \times @ + 10 \times @ + @)$
Simplifying the left side:
$100@ + 37 + 460 + @ + 500 + 10@ + 3 = (100@ + @ + 10@) + (37 + 460 + 500 + 3) = 111@ + 1000$
Simplifying the right side:
$1000 + 100@ + 10@ + @ = 1000 + 111@$
The equation becomes $111@ + 1000 = 111@ + 1000$.

This equation is an identity, which means it holds true for any value of @. Since @ represents a single digit in a number, it must be one of the digits from 0 to 9. Therefore, @ can be any digit. The structure of the column-wise addition also confirms this: in each column (units, tens, hundreds), the sum is $(10 + @)$, resulting in the units digit @ and a carry-over of 1 to the next column. The final carry-over of 1 becomes the thousands digit of the total sum, which is also 1, matching the format 1@@@@. This structure works regardless of the specific digit @, as long as it’s 0-9.

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