If x and yare positive integers , what is the remainder when

If x and y are positive integers, what is the remainder when 10^x +y is divided by 3?

First thing that comes to mind: What is the divisibility rule of 3?
Sum of all digits should be divisible by 3. What is the remainder when a number is divided by 3? It is same as the remainder you get when you divide the sum of the digits by 3. So, to get the remainder, all you need is the sum of the digits of the given number.

Given number: \(10^x + y\)

If x is a positive integer, what will \(10^x\) look like?
10 or 100 or 1000 or 10000 or 100000 etc. What is the sum of digits in each one of these cases? Of course 1.
Whatever y is, it gets added to this number. The sum of the digits of \(10^x + y\) will depend on the value of y.
Say y = 20. Sum of digits of \(10^x + y\) will be 1 (from before) + 2 = 3
Say y = 7. Sum of digits of \(10^x + y\) will be 1 (from before) + 7 = 8

You can find the sum of the digits the moment you get the value of y. Hence statement 2 is sufficient alone.

The question is not really understandable. Please post it either in the "formula form" or like this (10^x)+y.

10^x +y means \(10^x +y\), so no ambiguity there. If it were 10^(x +y) it would be written that way. Still edited the original post to avoid further confusions.

I'm sorry if this is something obvious i'm missing but I'm not quite following what you mean by that; sum of digits 10^x is always 1 as in...10 and x?

If x = 1, then 10^1 = 10 --> the sum of the digits = 1 + 0 = 1;
If x = 2, then 10^3 = 100 --> the sum of the digits = 1 + 0 + 0 = 1;
If x = 3, then 10^3 = 1000 --> the sum of the digits = 1 + 0 + 0 + 0= 1;
...

Hope it's clear.

Sorry I'm still struggling with this; what are you equating the 1 and the 0's to, the 1 being the 10 and the 0's being the power it is risen to?

We are talking about the sum of the digits of a number. For example, the sum of the digits of 17 is 8 because 1 + 7 = 8. Or, the sum of the digits of 2^5 is 5 because 2^5 = 32 and 3 + 2 = 5.

Hi,

INFO:-

since the div rule by 3 states that the sum of digits should be div by 3 and the remainder of any integer is same as remainder of the sum of integer..
lets see what is 10^x + y...
here irrespective of value of x, the sum of integers will be 1, as it will be 1,10,100,1000...
so we require to know y to find the remainder..

lets see the choices..
1) X=5
nothing about y
insuff

(2) y=2
now we know sum = 1+2..
remainder is 0..
Suff
B
Hope it helped

Solution:

Question Stem Analysis:

We need to determine the remainder when 10^x + y is divided by 3, given that x and y are positive integers. Notice that 10^x is the number 1 followed by x zeros. Since the sum of these digits is 1, the remainder when 10^x is divided by 3 will be 1. Therefore, the answer to the question depends on the value of y. For example, if y = 7, then the remainder will be 2 since the sum of the digits of 10^x + y will be 1 + 7 = 8, and 8 divided by 3 is 2, with a remainder of 2.

Statement One Alone:

Since we don’t know anything about y, statement one is not sufficient.

Statement Two Alone:

Since y = 2, the sum of the digits of 10^x + y will be 1 + 2 = 3, which has a remainder of 0 when divided by 3. Statement two is sufficient.

Answer: B

I didn't understand:
>Say y = 20. Sum of digits of
[ltr]10x+y10x+y[/ltr]
will be 1 (from before) + 2 = 3
Say y = 7. Sum of digits of
[ltr]10x+y10x+y[/ltr]
will be 1 (from before) + 7 = 8.
Can you please help me to understand this also is there another way to handle this question? @KarishmaB

What exactly is your question?

If y = 20, the sum of the digits of 10^x + y will be 3, making 10^x + y divisible by 3.

If y = 7, the sum of the digits of 10^x + y will be 8, so 10^x + y will give a remainder of 2 when divided by 3.

Analyzing the question from the sum of the digits standpoint is the best way to handle this question.

P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

• Word Problems
• Work Problems
• Distance Problems
• Mixture Problems
• Percent and Interest Problems
• Overlapping Sets Problems
• Statistics Problems
• Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this and similar questions.

Hope it helps.