Question

From a collection of eight cards numbered 1 to 8, if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is:

• A. \( \frac{3}{14} \)
• B. \( \frac{6}{13} \)
• C. \( \frac{3}{16} \)
• D. \( \frac{1}{4} \)

Hint:

When determining the probability of a perfect square, first check the prime factorizations of all possible pairs, and then count the number of favorable pairs.

Answer 1

The Correct Option is C

We are given a set of 8 cards numbered 1 to 8. We are drawing two cards at random, with replacement, and need to find the probability that the product of the numbers on the two cards is a perfect square. Step 1: The numbers on the cards are: \( 1, 2, 3, 4, 5, 6, 7, 8 \). A perfect square is a number whose prime factorization contains even powers of all primes. Step 2: The product of two numbers \( a \) and \( b \) is a perfect square if the prime factorization of \( a \times b \) contains even powers of all primes. Now, let's check all pairs of numbers \( a \) and \( b \) such that \( a \times b \) is a perfect square: - \( 1 \times 1 = 1 \) - \( 4 \times 4 = 16 \) - \( 9 \times 9 = 81 \) - \( 2 \times 2 = 4 \) Step 3: Count the number of pairs. The total number of pairs when drawing two cards with replacement is: \[ 8 \times 8 = 64 \] Step 4: There are three pairs where the product is a perfect square: \( (1, 1), (4, 4), (9, 9) \). Thus, the probability is: \[ P(\text{perfect square}) = \frac{3}{64} \] So, the correct answer is \( \frac{3}{16} \). % Final Answer \[ \boxed{\frac{3}{16}} \]