A prime number less than 100 is selected randomly. What is the probability that its square has units digit 7?
zero. No square ends in 7.
Only 1,4,5,6,9,0
Poke around a bit and you will quickly see why.
To find the probability that the square of a prime number less than 100 has a units digit of 7, we need to consider the set of prime numbers less than 100 and determine how many of those primes have a square with a units digit of 7.
Step 1: Find all prime numbers less than 100.
Prime numbers less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Step 2: Calculate the squares of the primes.
The squares of these prime numbers are: 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409.
Step 3: Count the number of squares that have a units digit of 7.
From the list above, we can see that there are 4 prime numbers (7, 17, 37, 47) whose squares have a units digit of 7.
Step 4: Calculate the probability.
The total number of prime numbers less than 100 is 25, and the number of those primes whose squares have a units digit of 7 is 4. Therefore, the probability is 4/25 or 0.16.
Thus, the probability that a randomly selected prime number less than 100 will have a square with a unit digit of 7 is 0.16 or 16%.
To find the probability that the square of a randomly selected prime number less than 100 has a units digit of 7, we need to consider two factors: the total number of prime numbers less than 100 and the number of primes whose square has a units digit of 7.
Step 1: Count the total number of prime numbers less than 100.
To find the total number of prime numbers less than 100, we can manually list them or use the Sieve of Eratosthenes method to identify them. By doing so, we can determine that there are 25 prime numbers less than 100.
Step 2: Count the number of primes whose square has a units digit of 7.
To find the prime numbers whose square has a units digit of 7, we can consider all the possible units digits for the prime numbers and determine if their squares end in 7. From 1 to 9, we can easily identify that the units digits creating a 7 when squared are 3, 7, and 1 (since 3^2 = 9, 7^2 = 49, and 1^2 = 1). However, numbers ending in 3 and 7 are multiples of 3 and therefore not prime. Thus, only the numbers ending in 1 are prime. Between 1 and 9, there is only one prime number that ends in 1, which is 1 itself.
Step 3: Calculate the probability.
To calculate the probability, we divide the number of primes whose square has a units digit of 7 by the total number of prime numbers less than 100.
Probability = Number of primes whose square has a units digit of 7 / Total number of prime numbers < 100
Probability = 1 / 25
Therefore, the probability that the square of a randomly selected prime number less than 100 has a units digit of 7 is 1/25.