Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:
(a) are divisible by 5 and by 7.
(b) have distinct digits.
(c) are not divisible by either 5 or 7.
(a) div. by 5 and by 7 => div. by 35
We can find out that
35*286=10010 and 35*29=1015
Therefore the number divisible by 35, n35
= 286-29=257
(b) distinct digits
First digit has 9 choices (1-9)
second and subsequent digits 9,8,7 choices each
Numbers with distinct digits
= 9*9*8*7
= 4536
(c) not divisible by either 5 or 7
Divisible by 5, n5= (10000-1000)/5=1800
Divisisble by 7, n7 = (10003-1001)/7=1286
Divisible by 5 or 7 or both
=n5+n7-n35
=1800+1286-257
=2829
Numbers NOT divisible by either 5 or 7
=(10000-1000)-2829
=6171
(a) For a number to be divisible by both 5 and 7, it needs to be divisible by their least common multiple, which is 35. Therefore, we need to find the number of positive integers between 1000 and 9999 that are divisible by 35.
To find this, we can divide 9999 by 35 and subtract the quotient of 1000 divided by 35, since we want to exclude any numbers below 1000.
9999 / 35 = 285 remainder 14
1000 / 35 = 28 remainder 20
So, there are 285 - 28 = 257 positive integers between 1000 and 9999 that are divisible by both 5 and 7.
(b) For the digits to be distinct, we need to count the number of permutations of four distinct digits. We can choose the first digit in 9 ways (0 is not included as a possible first digit since it would make the number a three-digit number), the second digit in 9 ways (we can't repeat the first digit), the third digit in 8 ways, and the fourth digit in 7 ways.
So, there are 9 * 9 * 8 * 7 = 4536 positive integers between 1000 and 9999 that have distinct digits.
(c) To find the numbers that are not divisible by either 5 or 7, we need to subtract the numbers from parts (a) and (b) from the total number of positive integers between 1000 and 9999.
The total number of positive integers between 1000 and 9999 is 9999 - 1000 + 1 = 9000.
So, the number of positive integers between 1000 and 9999 that are not divisible by either 5 or 7 is 9000 - 257 - 4536 = 4207.
Therefore, there are 4207 positive integers between 1000 and 9999 that have the properties (a), (b), and (c).
(a) To find the number of positive integers between 1000 and 9999 inclusive that are divisible by 5 and 7, we need to find the multiples of the least common multiple (LCM) of 5 and 7 within that range.
The LCM of 5 and 7 is 35. We need to find the multiples of 35 between 1000 and 9999.
Dividing 1000 by 35 gives us 28 remainder 20. This means that the first multiple of 35 greater than or equal to 1000 is 35 * 29 = 1015.
Dividing 9999 by 35 gives us 285 remainder 24. This means that the last multiple of 35 within the range is 35 * 285 = 9975.
To find the count of the multiples of 35 between 1015 and 9975 (inclusive), we can use the formula:
Count = (Last multiple - First multiple) / 35 + 1.
Count = (9975 - 1015)/35 + 1
Count = 8960/35 + 1
Count = 256 + 1
Count = 257.
Therefore, there are 257 positive integers between 1000 and 9999 inclusive that are divisible by both 5 and 7.
(b) To find the number of positive integers between 1000 and 9999 inclusive that have distinct digits, we need to consider the possibilities for each digit.
For the thousands digit, we have 9 choices (1-9, as 0 is not allowed).
For the hundreds digit, we have 9 choices (excluding the digit used for thousands).
For the tens digit, we have 8 choices (excluding the digits used for thousands and hundreds).
For the units digit, we have 7 choices (excluding the digits used for thousands, hundreds, and tens).
Therefore, the total number of positive integers with distinct digits is: 9 * 9 * 8 * 7 = 4,536.
(c) To find the number of positive integers between 1000 and 9999 inclusive that are not divisible by either 5 or 7, we need to find the total count of numbers within the range and subtract the count of numbers that are divisible by 5 or 7.
Total count of numbers within the range = 9999 - 1000 + 1 = 9000.
To find the count of numbers divisible by 5 or 7, we can add the count of numbers divisible by 5 and the count of numbers divisible by 7, and then subtract the count of numbers divisible by both 5 and 7 (as it was already included in both counts).
Count of numbers divisible by 5 = (9999 - 1015)/5 + 1 = 1807.
Count of numbers divisible by 7 = (9975 - 1022)/7 + 1 = 1422.
Count of numbers divisible by both 5 and 7 = (9975 - 1015)/35 + 1 = 257.
Count of numbers divisible by 5 or 7 = 1807 + 1422 - 257 = 1972.
Count of numbers not divisible by either 5 or 7 = Total count - Count of numbers divisible by 5 or 7
= 9000 - 1972
= 7028.
Therefore, there are 7028 positive integers between 1000 and 9999 inclusive that are not divisible by either 5 or 7.
To find the number of positive integers with exactly four decimal digits that satisfy the given conditions, we can break down the problem into three parts and solve each individually.
(a) The number should be divisible by both 5 and 7.
To find numbers that are divisible by 5, we know that the last digit must be either 0 or 5 since any number ending with 0 or 5 is divisible by 5. Therefore, there are 2 choices for the last digit.
To find numbers that are divisible by 7, we cannot use a simple criteria based on a single digit. Instead, we can use the divisibility rule for 7, which states that a number is divisible by 7 if and only if the difference between twice the unit digit and the remaining number is divisible by 7.
So, we can start by writing down a chart with potential last digits (0 and 5) and then checking which numbers will satisfy the divisibility rule for 7:
For 0 as the last digit: We can have 10 choices for the first digit (0-9), 10 choices for the second digit, and 9 choices for the third digit (as it cannot be the same as the first digit). This gives us a total of 10 * 10 * 9 = 900 possibilities.
For 5 as the last digit: We can have 9 choices for the first digit (1-9, since 0 will make it a 3-digit number), 10 choices for the second digit (0-9), and 8 choices for the third digit (as it cannot be the same as the first or second digit). This gives us a total of 9 * 10 * 8 = 720 possibilities.
Now, to find the numbers that are divisible by both 5 and 7, we need to find the common multiples of 900 and 720 (since the last three digits can be arranged in any order). We can find the least common multiple (LCM) of 900 and 720, which is 3,600. Since the last three digits can be arranged in any order, we have 6 permutations for each number. So, the total number of positive integers satisfying condition (a) is 3,600/6 = 600.
(b) The number should have distinct digits.
To ensure that the digits are distinct, we can choose the digits from the remaining pool after selecting the last digit for condition (a).
For 0 as the last digit: We have 9 choices for the first digit (1-9, since 0 will make it a 3-digit number), 9 choices for the second digit (as it cannot be the same as the first digit), and 8 choices for the third digit (as it cannot be the same as the first or second digit). This gives us a total of 9 * 9 * 8 = 648 possibilities.
For 5 as the last digit: We have 9 choices for the first digit (1-9, since 0 will make it a 3-digit number), 9 choices for the second digit (as it cannot be the same as the first digit), and 7 choices for the third digit (as it cannot be the same as the first or second digit). This gives us a total of 9 * 9 * 7 = 567 possibilities.
Again, we can find the common multiples of 648 and 567 (last three digits arranged in any order), which is 3,888. Dividing by 6 (number of permutations) gives us 648.
(c) The number should not be divisible by either 5 or 7.
To find the numbers that are not divisible by 5 or 7, we need to find the total numbers minus the numbers found in (a) and (b).
Total numbers between 1000 and 9999 (inclusive): 9999 - 1000 + 1 = 9000.
Numbers satisfying (a): 600.
Numbers satisfying (b): 648.
Therefore, the number of positive integers satisfying condition (c) is 9000 - 600 - 648 = 7752.
In conclusion, there are 600 positive integers with exactly four decimal digits that are divisible by both 5 and 7, have distinct digits, and are not divisible by either 5 or 7.