Number Theory question

[amar.doshi]

I saw this question on a forum for the IIM CAT prep and seemed interesting.

Q: If a is an integer, and the tens digit of a^2 is 7, what is the units digit?
Ans: 6

So the way I solved this was brute force by squaring numbers till I got 24-square = 576, hence assumed the rule applied to all subsequent squares that would have a 7 in the tens digit (didn't really bother computing further, just assumed there is no exception and moved on).

Is there an easy way to do this using algebra? What am I missing?

Also, just for kicks I looked at the the squares from 1 - 100 to see if there are any such other patterns. I notice that for all squares which have an "odd number" in the tens digit, the units digit is 6!

So if a DS question were posed (and I'm making it up) that:

If a is a positive integer, is the tens digit of a^2 odd?
I. the units digit of a is equal to 4 or 6
II. the units digit of a is equal to 3 or 5

What would be the best way to approach this and solve quickly, assuming not having knowledge of the fact beforehand? Would you approach it using algebra or number plugging? (with number plugging I would have a little less confidence since I wouldn't be completely sure there isn't an exception somewhere)

--------------------
For reference,
Numbers with tens digit of the square as 1 are: 4, 46, 54, 96
Numbers with tens digit of the square as 3 are: 6, 44, 56, 94
Numbers with tens digit of the square as 5 are: 16, 34, 66, 84
Numbers with tens digit of the square as 7 are: 24, 26, 74, 76
Numbers with tens digit of the square as 9 are: 14, 36, 64, 86

[amar.doshi]

Anyone / Instructors?

Is there an easy way to do this using algebra? What am I missing?

and

If a is a positive integer, is the tens digit of a^2 odd?
I. the units digit of a is equal to 4 or 6
II. the units digit of a is equal to 3 or 5

What would be the best way to approach this and solve quickly, assuming not having knowledge of the fact beforehand? Would you approach it using algebra or number plugging? (with number plugging I would have a little less confidence since I wouldn't be completely sure there isn't an exception somewhere)

[RonPurewal]

eh.

there are algebraic ways to do this. for instance, see here:
http://mathforum.org/library/drmath/view/63466.html
it's the first problem on the page.

there is no "easy" way to do this problem.

HOWEVER,

remember two things:

* THERE ARE NO STYLE POINTS:
it doesn't matter whether your solution is "ugly" or "elegant"; all that matters is whether it solves the problem at hand.
in your case, just finding a single example is CLEARLY the best way to solve the problem. if you have the perfect squares memorized (which you should - probably up to 25^2, so this one would just barely fall within that range), then you should be able to come up with this example fairly quickly.

* THE INDIAN CAT PROBLEMS CAN BE MUCH MORE TECHNICALLY DEMANDING THAN THE GMAT PROBLEMS:
i've seen materials for the indian CAT exam; the emphasis is totally different.
- the CAT exam is much more demanding than the gmat, at least in terms of technical knowledge.
BUT
- the GMAT problems are less conventional, and more "clever". they technically don't go beyond the purview of early-high-school math, but they find all sorts of clever work-arounds and connections between different topics.

in other words - you're going to get much farther with memorized techniques and facts on the CAT tests, but you're going to have to study shortcuts, connections, and "beat the test" techniques for the GMAT.

make sure that you stay in this state of mind. this problem is WAY too technically difficult for the gmat, so, since this is a gmat forum, the short answer is just "find an example and you're done."

hope that helps.