A teacher wants to divide 84 students into equal groups such that the number of students i… — State PSC Exam

A teacher wants to divide $84$ students into equal groups such that the number of students in each group is a prime factor of $84$ . What is the sum of all distinct prime numbers that can represent the size of these groups?

$12$ Answer
B
$14$
C
$7$
D
$42$

Answer

The sum of all distinct prime group sizes is

12

The prime factorization of

84

2^2 \times 3 \times 7

. The distinct prime factors are

2

3

, and

7

. Summing these distinct prime numbers yields

2 + 3 + 7 = 12

Step-by-Step Solution

Find the prime factorization of

84

84 = 2^2 \times 3^1 \times 7^1

Decomposing

84

into prime factors reveals all prime numbers that divide

84

exactly.

Identify the distinct prime factors of

84

The distinct prime factors are

2

3

, and

7

The question specifies distinct prime factors, so each prime base is listed once regardless of its exponent.

Calculate the sum of these distinct prime factors.

2 + 3 + 7 = 12

Adding the distinct prime factors gives the sum of all possible prime group sizes.

Key Concept

Prime Factorization and Distinct Prime Factors

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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