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Difficulty: HardClassification of Numbers

A secure digital vault requires a 2-digit numerical access code. The code is determined by calculating the value of the expression (P+Q)−(R×S)(P + Q) - (R \times S), where the variables are defined based on the mathematical classification of numbers:

* PP: The sum of all single-digit prime numbers.
* QQ: The product of the smallest positive composite number and the smallest odd prime number.
* RR: The total count of whole numbers strictly less than 1010 that are classified as neither prime nor composite.
* SS: The smallest positive integer nn for which the expression 12n\sqrt{12n} results in a rational number.

What is the final numerical access code?

Answer: 23

Answer

23
The correct calculation evaluates each subset definition perfectly: P = 17 (sum of primes 2, 3, 5, 7), Q = 12 (4 * 3), R = 2 (0 and 1 are neither prime nor composite), and S = 3 (making 12 * 3 = 36 a perfect square). Plugging these into the equation (17 + 12) - (2 * 3) yields exactly 23.

Step-by-Step Solution

1
Identify single-digit primes and sum them to find P.
P = 17
The single-digit primes are 2, 3, 5, and 7. Summing them yields 2 + 3 + 5 + 7 = 17.
2
Identify the smallest composite and smallest odd prime to find Q.
Q = 12
The smallest positive composite number is 4, and the smallest odd prime number is 3. Multiplying them yields 4 * 3 = 12.
3
Count whole numbers less than 10 that are neither prime nor composite to find R.
R = 2
The set of whole numbers strictly less than 10 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Only 0 and 1 fit the classification of being neither prime nor composite.
4
Find the smallest positive integer n making \sqrt{12n} rational.
S = 3
For the square root to be rational, 12n must be a perfect square. Since 12 factors to 2^2 * 3, the smallest integer n to pair the remaining 3 is 3.
5
Evaluate the final expression.
23
(P + Q) - (R * S) = (17 + 12) - (2 * 3) = 29 - 6 = 23.

Key Concept

Classification properties of primes, composites, whole numbers, and rational numbers.
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Topics

Classification of Numbers

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