Square (Some characteristics)

Squares (Some Characteristics)

The following was a question posted by (sorry, I missed out the name) in one of the group study. Similar questions do come frequently in different papers.  

What is the least number to be added to 3000 to make it a perfect square?
a 191
b 136
c 25
d 84
e none of these

The answer to the above question is 25. Both 25 and 136 when added to 3000 would make it a square. Hence, the least number is 25.

Now let us take a look at some of the characteristics of a square.

Any number ending in odd no of ‘0’s can never be a square (10, 1000, 10000, etc)

All numbers ending in 2, 3, 7 and 8 can never be a square.  ( 22, 33, 77, 88 )

A square is always a multiple of either 3 or 4, plus or minus 1.

15 is a multiple of 3 but not a square. If 1 is added it becomes 16 which is a square.

24 is a multiple of both 3 and 4 but not a square. If 1 is added then it becomes 25 which is a square.

49 is already a square but not a multiple of 3 or 4. If you subtract 1 it becomes 48 which is a multiple of 3 and 4.

The square of an odd number is always odd and the square of an even number is always even.

Digital Root

100 is a square. The sum of the digits of 100 when reduced to a single digit is 1.

400 is a square. The sum of the digits of 400 when reduced to a single digit is 4.

169 is a square. The sum of the digits of 169 when reduced to a single digit is 7.

900 is a square. The sum of the digits of 900 when reduced to a single digit is 9.

The single digits 1, 4, 7 and 9 are called the digital roots. Please note, that any square number when reduced to a single digit as above will always be 1, 4, 7 or 9 and nothing else.

Now take the question above. 

Option (a) 191 and option (d) 84 are automatically out since when added to 3000 the reduced single digit will be other than 1, 4, 7 and 9.

Both options (b) 136 and option (c) 25 when added to 3000 makes a square and fulfils the rule of Digital Root. Between the two, 25 is the smallest number and hence is the answer.

Sometimes, when such additions or deletions made to the given number you may have two or more numbers fulfilling the condition of single digit being 1, 4, 7 or 9. There can only be one answer. In such cases when you actually add or subtract you may find, that out of the more than one possibility only one will fit as the answer and the others will fall under the category of numbers ending in 2, 3, 7 or 8.

148 single digit 4. But not a square since the number is ending in 8.

632 single digit 2. But not a square since the number is ending in 2.

733 single digit 4. But not a square since the number is ending in 3.

877 single digit 4. But not a square since the number is ending in 7.

You can find many such numbers during practice and once you understand the principles of this, answering questions of the above type will become easy.

I hope you find the above useful. For any clarification please feel free to write to me.