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by sanju09 » Fri Mar 05, 2010 6:21 am
Which of the following is/are TRUE?

I. If a number has n digits, then its square will have 2 n or 2 n - 1 digit(s).
II. A square number can always be written in the form 3 n or 3 n + 1, where n is any non-negative integer.
III. A square number can always be written in the form 4 n or 4 n + 1, where n is any non-negative integer.

(A) I only
(B) II & III only
(C) I & III only
(D) I & II only
(E) I, II, & III
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by yeahdisk » Fri Mar 05, 2010 7:34 am
I would go for C although I'm not sure of an algebraic proof? Is there one?
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by fibbonnaci » Fri Mar 05, 2010 7:49 am
I tried to solve this question by picking numbers.

I. lets say number - 5 (n= 1) 25 (n= 2)
- 13 (n=2) 169 (n=3) => (2n-1 digits)
Satisfies

II. lets consider the same numbers. 25 can be written as 3*8 +1
Similarly even 169 can be written in the form of 3n +1. You can check for any square number, you will see it satisfies this criteria.

III. say 25 can be written as 4*6 + 1
similarly you can check for other square numbers also. You will find it satisfies the criteria.

So my take is E
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by yeahdisk » Fri Mar 05, 2010 7:59 am
I think I agree with you
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by sharuvman » Fri Mar 05, 2010 9:24 am
II. would fail if n =2,
3n+1 = 7 which is not a square number

III. would fail if n = 1,
4n+1 = 5 which is again not a square number

I. Is always true
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by camilaross » Fri Mar 05, 2010 11:37 am
I'm not sure if I did something wrong.. but I tested with 5 and II and III were proven incorrect.
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by fibbonnaci » Fri Mar 05, 2010 11:40 am
note the term 'n' in I is different from II and III
In I, 'n' denotes the number of digits, which is not the case in II and III.
Also in II, we are talking about squared numbers. 5 is not a square number. 25 is a square of number 5, so we can call 25 to be square number

Hope this helps!
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by camilaross » Fri Mar 05, 2010 11:57 am
yeah.. I see it.... thanks!
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by kevincanspain » Fri Mar 05, 2010 12:33 pm
Note that each integer can be written as 2k or 2k + 1


Also, each integer can be written as 3k - 1 , 3k, or 3k + 1
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