Question

How many positive integers less than or equal to 1000 are divisible by none of 3, 8, and 25?

a. 136
b. 423
c. 662
d. 789

Answer 1

To find the number of positive integers less than or equal to 1000 that are not divisible by 3, 8, and 25, we can use the principle of inclusion-exclusion. The answer is 789.

Step-by-step explanation:

To find the number of positive integers less than or equal to 1000 that are not divisible by 3, 8, and 25, we can use the principle of inclusion-exclusion.

Step 1: Find the number of positive integers divisible by 3, 8, or 25.

• Divisible by 3: There are 1000/3 = 333 positive integers divisible by 3.
• Divisible by 8: There are 1000/8 = 125 positive integers divisible by 8.
• Divisible by 25: There are 1000/25 = 40 positive integers divisible by 25.

Step 2: Find the number of positive integers divisible by both 3 and 8, both 3 and 25, and both 8 and 25.

• Divisible by both 3 and 8: There are 1000/(3*8) = 41 positive integers divisible by both 3 and 8.
• Divisible by both 3 and 25: There are 1000/(3*25) = 13 positive integers divisible by both 3 and 25.
• Divisible by both 8 and 25: There are 1000/(8*25) = 5 positive integers divisible by both 8 and 25.

Step 3: Find the number of positive integers divisible by 3, 8, and 25.

• Divisible by 3, 8, and 25: There is 1000/(3*8*25) = 1 positive integer divisible by 3, 8, and 25.

Step 4: Apply the principle of inclusion-exclusion to find the number of positive integers divisible by none of 3, 8, and 25.

Number of positive integers divisible by none of 3, 8, and 25 = 1000 - (333 + 125 + 40 - 41 - 13 - 5 + 1)

= 789.