GMAT Math : Properties of Integers
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Example Questions
Example Question #41 : Arithmetic
A whole number has four digits. True or false: The integer is divisible by 12.
Statement 1: The sum of the digits in the number is 36.
Statement 2: The last digit of the number is 9.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. If the sum of the digits of a four-digit number is 36, then the average value of each digit is 
Now assume Statement 2 alone. For a number to be divisible by 12, it must also be divisible by any factors of 12, including 2. This is only possible if the last digit is even, which it is not. Therefore, the number is not divisible by 12.
Example Question #41 : Arithmetic
A whole number has four digits. True or false: The integer is divisible by 11.
Statement 1: The sum of the digits is 22.
Statement 2: The last two digits are 55.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Assume both statements to be true. The numbers 6,655 and 7,555 both end in 55 and have digit sum 22. However,
and
Only the former is divisible by 11, so the two statements together do not answer the question.
Example Question #43 : Arithmetic
True or false: Positive integer 
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. Then:
Either
Each of 5 and 7 has exactly two factors, 1 and itself, so either way 
Assume Statement 2 alone. Then:
Either
or
Each of 7 and 11 has exactly two factors, 1 and itself, so either way 
Example Question #44 : Arithmetic
True or false: Positive integer 
Statement 1: The last digit of 
Statement 2: 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Assume Statement 1 alone. An integer is divisible by 2 if and only if its last digit is 0, 2, 4, 6, or 8. Therefore, it therefore follows that 
Assume Statement 2 alone. Then:
Either
Therefore, it is not clear whether or not 
Example Question #45 : Dsq: Understanding The Properties Of Integers
A whole number has four digits. True or false: The integer is divisible by 4.
Statement 1: The last digit of the integer is 4.
Statement 2: The sum of the four digits is 20.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Consider the two four-digit whole numbers 5,564 and 5,474. Each ends in 4, and the sum of the digits in each integer is 20, satisfying the conditions of both statements. However, only the first integer is divisible by 4:
The two statements together are therefore insuffcient to answer the question.
Example Question #42 : Arithmetic
True or false: A positive integer 
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Assume both statements to be true.
From Statement 1:
Either
From Statement 2:
Either
or
Both equations have the same two solutions, 9 and 11, so it is unclear whether 
A prime number has exactly two factors, 1 and itself. 9 has 3 as a factor, so 9 is not prime; 11 has only 1 and 11 as factors, so 11 is prime. Since either 
Example Question #47 : Dsq: Understanding The Properties Of Integers
Statement 1:
Statement 2: 
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Assume Statement 1 alone. Then:
Either
However, it is given in the problem that 
Assume Statement 2 alone. 3 has only two factors, 1, which is not considered a prime number, and 3, which, having only two factors, is a prime number. Either 
Example Question #48 : Dsq: Understanding The Properties Of Integers
A whole number has four digits. True or false: The integer is divisible by 6.
Statement 1: The last digit in the number is 6.
Statement 2: The sum of the digits in the number is 24.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
A whole number is divisible by 6 if and only if it is divisible by all of the factors of 6, which, other than 1 and 6 itself, are 2 and 3. Therefore, it must be established that the number is divisible by both of these numbers. To be divisible by 2, it is necessary and sufficient that the last digit of the number be 0, 2, 4, 6, or 8. To be divisible by 3, it is necessary and sufficient that the sum of the digits be divisible by 3.
Statement 1 alone, therefore, proves that the number is divisible by 2; however, it does not address divisibility by 3. Similarly, Statement 2 alone proves that the number is divisible by 3 (since 24 is as well), but it does not address divisibility by 2. The two statements together, however, address divisibility by 6, since if both are true, it holds that the number is divisible by both 2 and 3, and, consequently, by 6.
Example Question #42 : Arithmetic
A whole number has four digits. True or false: The integer is divisible by 5.
Statement 1: The last digit is 5.
Statement 2: The sum of the digits is 25.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Assume Statement 1 alone. A necessary and sufficient condition for a whole number to be divisible by 5 is that its last digit be either 0 or 5. By Statement 1, the number meets this requirement, so it is divisible by 5. Statement 2, which gives the digit sum, is therefore, irrelevant.
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