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A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?
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A number is selected at random from the first 1000 natural numbers. Wh...
Probability of selecting a multiple of 7 or 11 from the first 1000 natural numbers

To solve this problem, we need to first find out how many numbers from 1 to 1000 are multiples of 7 or 11.

Finding the number of multiples of 7 or 11 in the first 1000 natural numbers

Multiples of 7: There are 142 multiples of 7 in the first 1000 natural numbers. This can be found by dividing 1000 by 7, which gives 142 with a remainder of 6. Therefore, there are 142 multiples of 7 in the first 1000 natural numbers.

Multiples of 11: There are 90 multiples of 11 in the first 1000 natural numbers. This can be found by dividing 1000 by 11, which gives 90 with a remainder of 10. Therefore, there are 90 multiples of 11 in the first 1000 natural numbers.

Finding the number of common multiples of 7 and 11 in the first 1000 natural numbers

Multiples of 7 and 11: To find the number of common multiples of 7 and 11 in the first 1000 natural numbers, we need to find the multiples of their LCM (Least Common Multiple), which is 77. There are 12 multiples of 77 in the first 1000 natural numbers. This can be found by dividing 1000 by 77, which gives 12 with a remainder of 44. Therefore, there are 12 common multiples of 7 and 11 in the first 1000 natural numbers.

Calculating the probability of selecting a multiple of 7 or 11 from the first 1000 natural numbers

Now that we know the number of multiples of 7 or 11 and the number of common multiples of 7 and 11 in the first 1000 natural numbers, we can calculate the probability of selecting a multiple of 7 or 11.

Probability = (Number of multiples of 7 or 11)/(Total number of natural numbers from 1 to 1000)

Total number of natural numbers from 1 to 1000 = 1000

Number of multiples of 7 or 11 = (142 + 90) - 12 = 220

Therefore, Probability = 220/1000 = 0.22 or 22%

Conclusion

The probability of selecting a multiple of 7 or 11 from the first 1000 natural numbers is 0.22 or 22%.
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A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?
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