Question

If $x$ and $y$ are prime numbers, is $y(x-3)$ odd? (1) $x>10$ (2) $y<3$

Short answer

No, y(x-3) is always even.

Step-by-step solution

Understand the properties of prime numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, etc. Notice that 2 is the only even prime number.

Analyze Statement (1): x > 10

Since x is a prime number greater than 10, it cannot be 2 (the only even prime). Therefore, x must be odd because any prime number other than 2 is odd.

Examine the expression y(x-3) when x is odd

For any odd prime number x, x-3 is also even (odd-odd=even). Therefore, y(x-3) = y(even number).

Analyze Statement (2): y < 3

Since y is a prime number less than 3, the only possible value for y is 2 (the only even prime number).

Evaluate y(x-3)

Substituting y = 2 into y(x-3), the expression becomes 2(even number), which is always even because any number multiplied by 2 is even.

Conclusion

Combining both statements, y(x-3) is always even.

Key concepts

Prime Numbers

Prime numbers are fundamental in mathematics. They are natural numbers greater than 1 that cannot be formed by multiplying two smaller natural numbers. A few examples are 2, 3, 5, 7, 11, and so on.
Notice, 2 is the smallest and the only even prime number. Every other prime is an odd number.
For example, if we consider primes larger than 2, they are all odd: 3, 5, 7, etc.
This property plays an essential role in problem-solving, specifically when working with expressions involving prime numbers.

Odd and Even Properties

Odd and even properties help in simplifying mathematical problems by understanding how numbers interact when added or multiplied.
For instance:

• Adding or subtracting odd numbers always yields an even result: odd + odd = even, odd - odd = even.
• Multiplying an odd number by an even number always yields an even result: odd * even = even.

These properties are pivotal to solving problems like the GMAT exercise mentioned. Identifying the parity (odd or even) of numbers in expressions can help in determining the final result. For example, since all primes greater than 2 are odd, subtracting 3 from any prime greater than 2 will result in an even number.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and the ability to interpret and solve problems using mathematical concepts.
In the given problem:
To check if the expression y(x-3) is odd, we need to reason through the properties of prime numbers and their interaction in arithmetic operations.
We know from the properties of prime numbers and the odd/even logic that:

• Since x is a prime number greater than 10, it must be odd.
• Thus, x - 3 is even (odd - odd = even).
• When y is a prime number less than 3, y must be 2 (the only even prime).
• So, y(even number) = 2(even number), which results in an even number.

Using these logical steps helps in clearly understanding and solving the problem without confusion.

Data Sufficiency

Data Sufficiency is a critical part of problem-solving where you determine if the given information is enough to solve a problem.
This type of problem appears frequently in GMAT exams.
For the given problem:
We need to check if statements (1) and (2) provide sufficient data to determine if y(x-3) is odd.

• Statement (1): Knowing x > 10, we deduce x must be odd, making x - 3 even.
• Statement (2): Knowing y < 3, y has to be 2 (since 2 is the only even prime below 3).

By combining both statements, we conclude y * (x - 3) is even.
Thus, the data given is sufficient to determine the outcome.
This structured approach is essential to mastering data sufficiency problems.