How many 4-digit numbers are neither multiples of 2 nor multiples of 5?
smallest 4 digit number: 1000
largest 4 digit number : 9999
number of 4 digit number = 9999-1000+1 = 9000
of those 4500 will be divisible by 2
How many are divisible by 5? ...> 9000/5 or 1800
how many are divisible by both 2 and 5 ? , they have to end in 0
there are 9000/10 = 900 , we can't count those twice
number divisible by 2 or 5 = 4500 + 1800 - 900 = 5400
so, those not divisible by either 2 or 5 = 9000 - 5400 = 3600
thanks
Ah, the perfect question for a math-loving clown like me! So, let's break it down. We have a total of four digits to work with, meaning we have a range of 0 to 9 for each digit.
To find the number of 4-digit numbers that are multiples of 2, we need to count how many numbers in the range of 1000 to 9999 are divisible by 2. Now, half of the numbers in this range will be multiples of 2, so we have (9999 - 1000 + 1) / 2 = 4500 multiples of 2 in total.
Similarly, to find the number of 4-digit numbers that are multiples of 5, we divide the number of integers in the range of 1000 to 9999 by 5. Again, we have (9999 - 1000 + 1) / 5 = 1800 multiples of 5.
Now, the trick is to find the intersection of these sets of multiples of 2 and multiples of 5. To do that, we want to find the common multiples. Since both 2 and 5 are prime numbers, their least common multiple is 2 * 5 = 10.
So, we divide the total number of multiples of 10 in the range of 1000 to 9999, which is (9999 - 1000 + 1) / 10 = 900.
Finally, to find the number of 4-digit numbers that are neither multiples of 2 nor multiples of 5, we subtract the number of multiples of 10 from the total number of 4-digit numbers in the given range:
Total 4-digit numbers = 9999 - 1000 + 1 = 9000.
Therefore, the number of 4-digit numbers that are neither multiples of 2 nor multiples of 5 is 9000 - 900 = 8100.
And that's no joke, my friend!
To find the number of 4-digit numbers that are neither multiples of 2 nor multiples of 5, we need to subtract the numbers that are multiples of both 2 and 5 from the total number of 4-digit numbers.
Step 1: Total 4-digit numbers:
The first 4-digit number is 1000 and the last 4-digit number is 9999.
So, the total number of 4-digit numbers is: 9999 - 1000 + 1 = 9000.
Step 2: Number of 4-digit numbers that are multiples of 2:
To be a multiple of 2, the last digit must be an even number (i.e., 0, 2, 4, 6, or 8).
So, there are 5 choices for the last digit, and for each of the remaining three digits, there are 10 choices (0 to 9).
Thus, the number of 4-digit numbers that are multiples of 2 is: 5 * 10 * 10 * 10 = 5000.
Step 3: Number of 4-digit numbers that are multiples of 5:
To be a multiple of 5, the last digit must be either 0 or 5.
So, there are 2 choices for the last digit, and for each of the remaining three digits, there are 10 choices.
Thus, the number of 4-digit numbers that are multiples of 5 is: 2 * 10 * 10 * 10 = 2000.
Step 4: Number of 4-digit numbers that are multiples of both 2 and 5 (i.e., multiples of 10):
To be a multiple of 10, the last digit must be 0.
So, there is 1 choice for the last digit, and for each of the remaining three digits, there are 10 choices.
Thus, the number of 4-digit numbers that are multiples of both 2 and 5 is: 1 * 10 * 10 * 10 = 1000.
Step 5: Calculate the number of 4-digit numbers that are neither multiples of 2 nor multiples of 5:
To get this number, we subtract the number of numbers that are multiples of both 2 and 5 from the total number of 4-digit numbers:
9000 - 1000 = 8000.
Therefore, there are 8000 4-digit numbers that are neither multiples of 2 nor multiples of 5.
To find the number of 4-digit numbers that are neither multiples of 2 nor multiples of 5, we can use the principle of inclusion-exclusion.
Step 1: Counting the multiples of 2
A multiple of 2 is a number that can be divided evenly by 2. Since we are looking for 4-digit numbers, the first 4-digit multiple of 2 is 1000. The last 4-digit multiple of 2 is 9998. To count the number of multiples of 2, we need to find how many integers between 1000 and 9998 (inclusive) are divisible by 2.
The largest even number less than 9998 is 9998 - 2 = 9996. The smallest even number greater than or equal to 1000 is 1000 + 2 = 1002.
To find the number of even numbers between 1002 and 9996, we subtract the two values and add 1: (9996 - 1002) + 1 = 8995.
Step 2: Counting the multiples of 5
A multiple of 5 is a number that can be divided evenly by 5. The first 4-digit multiple of 5 is 1000. The last 4-digit multiple of 5 is 9995. To count the number of multiples of 5, we need to find how many integers between 1000 and 9995 (inclusive) are divisible by 5.
The largest multiple of 5 less than 9995 is 9995 - 5 = 9990. The smallest multiple of 5 greater than or equal to 1000 is 1000 + 5 = 1005.
To find the number of multiples of 5 between 1005 and 9990, we subtract the two values and add 1: (9990 - 1005) + 1 = 8986.
Step 3: Applying the inclusion-exclusion principle
To find the count of numbers that are neither multiples of 2 nor multiples of 5, we subtract the numbers counted in Step 1 and Step 2 from the total count of 4-digit numbers (9999 - 1000) + 1.
Total count of 4-digit numbers = (9999 - 1000) + 1 = 9000
Count of numbers divisible by 2 = 8995
Count of numbers divisible by 5 = 8986
Count of numbers that are multiples of 2 and 5 (divisible by 10) = Count of numbers divisible by 10 = 8990
Count of numbers neither divisible by 2 nor divisible by 5 = Total count of 4-digit numbers - (Count of numbers divisible by 2 + Count of numbers divisible by 5 - Count of numbers divisible by 10) = 9000 - (8995 + 8986 - 8990) = 9000 - 17991 = 72.
Therefore, there are 72 4-digit numbers that are neither multiples of 2 nor multiples of 5.