Consider the set of the first 100 positive integers (from 1 to 100 inclusive). An integer … — State PSC Exam

Consider the set of the first 100 positive integers (from 1 to 100 inclusive).

An integer $N$ from this set satisfies all of the following three conditions simultaneously:
1. $N$ is a composite number.
2. $N$ is neither divisible by 2 nor divisible by 3.
3. The square root of $N$ is an irrational number.

What is the total number of possible values for $N$ ?

Answer: 7

Answer

There are exactly 7 values for N that satisfy all three conditions.

The complete set of numbers not divisible by 2 or 3 from 1 to 100 contains 33 integers. Excluding the number 1 (neither prime nor composite) and the 23 prime numbers leaves exactly 9 composite numbers. From these 9 composites, 25 and 49 must be excluded because their square roots are 5 and 7 (rational numbers). This leaves 7 valid integers.

Step-by-Step Solution

Determine the total number of integers from 1 to 100 that are neither divisible by 2 nor divisible by 3.

There are 33 such numbers.

Using the inclusion-exclusion principle: there are 50 multiples of 2, 33 multiples of 3, and 16 multiples of 6. Multiples of 2 or 3 = 50 + 33 - 16 = 67. The remaining numbers are 100 - 67 = 33.

Filter the 33 remaining integers to find those that are composite.

There are 9 composite numbers: 25, 35, 49, 55, 65, 77, 85, 91, and 95.

Of the 33 numbers, 1 is neither prime nor composite. There are 25 primes up to 100, and excluding 2 and 3 leaves 23 primes. Thus, the composites are 33 - 1 - 23 = 9. These are the products of primes 5 and greater.

Eliminate numbers from the composite list whose square roots are rational.

Remove 25 and 49.

Condition 3 requires the square root of N to be irrational. An integer has a rational square root if and only if it is a perfect square. In our list, 25 and 49 are perfect squares.

Count the final remaining valid numbers.

7 numbers remain.

Subtracting the 2 perfect squares from the 9 composite numbers leaves 7 numbers that satisfy all three conditions.

Key Concept

Classification of numbers combining prime/composite definitions, divisibility principles, and properties of rational and irrational roots.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic