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# A five-digit number is chosen at random from all possible five-digit numbers.

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A five-digit number is chosen at random from all possible five-digit numbers. Find the probability that the number is NOT divisible by every element of (1,2,3,4,5,6)

Feb 4, 2023

#1
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The number is divisible by 6 if and only if it is divisible by both 2 and 3. To find the total number of 5-digit numbers, we consider all numbers between 10,000 and 99,999, inclusive, which is 90,000. To find the number of numbers that are not divisible by any of (1,2,3,4,5,6), we can find the number of numbers that are divisible by each of these factors and subtract this number from 90,000.

The number of numbers that are divisible by 2 is 45,000, the number that are divisible by 3 is 30,000, and the number that are divisible by 6 is 15,000. The number of numbers divisible by 4, 5, and 6 can be calculated in a similar fashion.

To find the number of numbers divisible by the least common multiple (LCM) of (2, 3, 4, 5, 6), we find the LCM and then find the number of 5-digit numbers divisible by this LCM. The LCM of (2, 3, 4, 5, 6) is 60, and there are 15,000 numbers divisible by 60.

Therefore, the number of numbers that are divisible by at least one of the numbers in the set (1,2,3,4,5,6) is 45,000 + 30,000 + 15,000 - 15,000 = 60,000.

So, the number of numbers that are not divisible by any of (1,2,3,4,5,6) is 90,000 - 60,000 = 30,000.

Finally, the probability of a random 5-digit number being NOT divisible by any of the numbers in the set (1,2,3,4,5,6) is 30,000/90,000 = 1/3.

Feb 4, 2023
#2
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The LCM of [1,2,3,4,5,6] is going to be 60. The least 5 digit number that is a multiple of 60 is going to be 10020 = 60 x 170, the greatest being 99960 = 60 x 1666. from 170 to 1666, there will be 1666-170+1 multiples of 60, so 1497 multiples. There is a total of 99999-10000+1 = 90000 5 digit numbers so the odds of a 5 digit number not being divisible by all of the numbers in the set is going to be $$\frac{90000-1497}{90000}$$ which is going to be

NOTE:Guest, the problem says that the number is not divisible by EVERY ELEMENT IN THE SET

$$\frac{29501}{30000}$$

Feb 4, 2023
edited by Imcool  Feb 4, 2023
#3
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The total number of 5-digit numbers is 90,000 (from 10,000 to 99,999).
Out of these, numbers divisible by 2, 3, 4, 5, 6 are 30,000, 20,000, 15,000, 12,000, 10,000 respectively.

The number of 5-digit numbers divisible by both 2 and 3 is 6000, and those divisible by 2, 3, and 4 are 2000. The number divisible by 2, 3, 4, 5 is 800, and those divisible by 2, 3, 4, 5, 6 is 400.

Therefore, the number of 5-digit numbers divisible by (1, 2, 3, 4, 5, 6) is 400.
Hence, the probability that the number is NOT divisible by every element of (1,2,3,4,5,6) is (90,000-400)/90,000 = 89,600/90,000 = 224/225.

Feb 4, 2023
#4
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Note: Make sure there are no duplications in your counting:

A short computer code shows there are: 24,000 5-digit integers that are NOT divisible by all the elements in the set {1, 2, 3, 4, 5, 6}. They begin like this:

(10001 , 10003 , 10007 , 10009 , 10013 , 10019 , 10021 , 10027 , 10031 , 10033 , 10037 , 10039.......and end like this:

99953 , 99959 , 99961 , 99967 , 99971 , 99973 , 99977 , 99979 , 99983 , 99989 , 99991 , 99997)

Therefore, the probability is: 24,000 / 90,000 ==4 / 15

Feb 4, 2023