Hey answer is pretty simple. !!
there are few rules for division you need to understand before answering such kind of questions.
Rule I. divisible by "2" - any no. will be considered divisible by two if the last digit of that no. is divisible by 2.

ex XXX4 - this no. will always be divisible by 2

Rule II. for numbers "3" and "9"

if sum of the digits of number is divisible by "3" or "9" then the number is considered divisible by "3" or "9"
consider no. 123456 , sum of digits = 1+2+3+4+5+6=21 , now 21 is divisible by 3 only not by 9. therefore 123456 is divisible by 3 only.

Rule III divisible by "4"
for any no. to be divisible by 4 , its last two digits should be divisible by 4.
eg. X32 will always be divisible by 4 for any value of X. You can try this exercise yourself

Sum of digits should be divisible by 9 in order for a number to be divisible by 9.... so add the digits.

A by applying divisibility rules. It is difficult to remember the rules though.

A small tip: Divisibility rule for 2^n - Last n digits should be divisible by the number

For example
for 2 - check last 1 digit
for 4 - check last 2 digits
for 8 - check last 3 digits
etc.

Rules are like following:

2: last digit should be divisible by 2.
3: sum of the digits should be divisible by 3.
4: last 2 digits should be divisible by 4.
5: last digit should be 0 or 5.
6: last digit should be divisible by 2 and sum of the digits should be divisible by 3.
7: given below
8: last 3 digits should be divisible by 8.
9: sum of the digits should be divisible by 9.

Divisibility by 7:-
Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example: 287 (Double the 7 to get 14. Subtract 14 from 28 to get 7 which is divisible by 7 and we can now say that 287 is divisible by 7.




Among these 7 is a bit tricky.
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My debrief: done-and-dusted-730-q49-v40

A = 1543&2. What is &?

1 - A is divisible by 9. That means 1+5+4+3+2+& must be divisible by 9. Now 1+5+4+3+2 is 15. Although, in general & can any integer value from 0 - 9, but it can only take a value that makes A a multile of 9.
When &=3, A = 18 --> A is divisible by 9
we need 9 more to reach next multiple of 9, but since we have already used 3 to reach 18 (15 + 3 = 18), we need 12 to reach next multiple of 9, i.e 27. (15 + 12 = 27)
Because & = 12 is not possible, we can only use & = 3.
Statement 1 is sufficient.

2 - For a number to be divisible by 4, its last two digits must be divisible by 4 (i.e a multiple of 4). Considering this rule, 154312 should be divisible by 4. 154352 should also be divisible by 4. Because we have found two values for &, this statement is not sufficient.

Thus answer is A.

Hope my explanations helps...

Divisibility Rules

2 - If the last digit is even, the number is divisible by 2.

3 - If the sum of the digits is divisible by 3, the number is also.

4 - If the last two digits form a number divisible by 4, the number is also.

5 - If the last digit is a 5 or a 0, the number is divisible by 5.

6 - If the number is divisible by both 3 and 2, it is also divisible by 6.

7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

8 - If the last three digits of a number are divisible by 8, then so is the whole number.

9 - If the sum of the digits is divisible by 9, so is the number.

10 - If the number ends in 0, it is divisible by 10.

11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.
Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.

12 - If the number is divisible by both 3 and 4, it is also divisible by 12.

25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.

For more on Number Theory check this: math-number-theory-88376.html

Hope it helps.
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@ represents the tens digit in integer A = 1543@2. What is the value of @ ?

(1) A is divisible by 9. In order an integer to be divisible by 9, the sum of its digit must be divisible by 9. Hence 1+5+4+3+@+2=15+@ must be divisible by 9, and since @ is a digit then it can only equal to 3. Sufficient.

(2) A is divisible by 4. In order an integer to be divisible by 4, its last two digits must be divisible by 4. Hence @2 must be divisible by 4, which is possible if @=1, @=3, @=5, @=7, or @=9. Not sufficient.

Answer: A.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


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