It's Prime Time - The Solutions
The smallest two-digit prime is 11, and the largest one-digit prime is 7. Their product is 77.
The square root of 143 is about 11.9, so any factor pair will include an integer less than or equal to 11. Therefore, we only need to check the prime numbers less than or equal to 11. These are 2, 3, 5, 7 and 11 so we would have to rule out only 5 divisors in order to declare 143 a prime number.
The prime digits are 2, 3, 5 and 7. Since we are limited to integers greater than 50,000, the ten-thousands digit can be either a 5 or 7. The thousands digit can be any of the four prime digits, as can the hundreds and tens digit. However, the units digit must be the 2 since the five-digit integer is even. Using the Counting Principle, we have that there are 2 × 4 × 4 × 4 × 1 = 128 integers that satisfy the conditions.