ankur.agrawal wrote:If (243)^x * (463)^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7

(2) x = 4

OA after some discussion.

Interesting question!

As always, one key to success is to understand exactly what the question is asking. Here, we only care about the units (i.e. ones) digit of n; accordingly, we only care about the units digit of (243^x)*(463^y).

Since both terms end in "3", we're multiplying (x+y) numbers that end in 3. In effect, we could simplify the question to:

What's the units digit of (3^x)*(3^y)?

or, even simpler:

What's the units digit of 3^(x+y)?

Consequently, to determine the units digit of the product, we need to know the value of (x+y).

(1) exactly what we're looking for! Sufficient.

(2) no info about y, so insufficient.

(1) is sufficient, (2) isn't: choose (A)!

Hi IMO- A

Actually its all about 3.

since both the figures are ending in 3 we actually are calculating 3^x+y =n

statement 1 gives value of x+y hence sufficient
statement 2 gives only x

Yes dear Stuart this is what i just posted. We think alike bro...


Thanks anyways.

Thanks guys

Wait a minute guys, as i see twist in the tail!!!!According to me, it is not just about 3 but knowing exactly which value would a unit digit take. I set out my approach below,
The question asks what is the unit digit of n when 243^x * 463^y = n
A] x+y = 7
possible scenarios for unit digit
x y 3^x 3^y Unit digit of the product
1 6 3 9 7
2 5 9 1 9
3 4 7 1 7

repeat as x&y interchange
Now there is no one value for the unit digit so A is ruled out
Choice left BCE

B] x=4 not clear as y not stated
B is out
combine A&B and you get one answer
So answer is C

eccentric wrote:Wait a minute guys, as i see twist in the tail!!!!According to me, it is not just about 3 but knowing exactly which value would a unit digit take. I set out my approach below,
The question asks what is the unit digit of n when 243^x * 463^y = n
A] x+y = 7
possible scenarios for unit digit
x y 3^x 3^y Unit digit of the product
1 6 3 9 7
2 5 9 1 9
3 4 7 1 7

repeat as x&y interchange
Now there is no one value for the unit digit so A is ruled out
Choice left BCE

I'm really not sure where you came up with that pattern, but it definitely doesn't match what will happen on this question.

If you take ANY 7 numbers ending in 3 and multiply them together, the units digit of the full product will be the same as the units digit of 3^7.

Now, powers of 3 definitely have a cycle to their units digits:

3, 9, 27, 81, 243, ..9, ...7, ...1, and so on...

So, for powers of 3, the units digit will be one of the 4 numbers {3, 9, 7, 1}.

Since we're multiplying 7 "3"s, our units digit will be 7, regardless of the specific values of x and y.

Thanks Stuart, your explanation lead me to revisit my pattern{(0,7),(1,6),(2,5),(3,4),(4,3),...} and was realized my silly mistake. yes A is sufficient as unit value for each pattern is 7..