Respuesta :
Answer:
(a) 1000
(b) 4500
(c) 4536
(d) 3000
(e) 2829
(f) 6171
(g) 1543
(h) 257
Step-by-step explanation:
Let X be positive integers between 1000 and 9999 which contains 9000 integers.
(a)The integers divided by the number of elements 9 is:
Use the quotient rule = absolute number of integers(X)/ number of elements
i = |9000|/9 = 1000
(b) Similar we use the quotient rule but we use an even number since even numbers are divided by 2 we can use 2.
i = |9000|/2 = 4500
(c) There are 10 digits
The first digit cannot be be zero so you can divide is 9 ways
The second digit can be zero but not the same as the first digit and therefore 9 ways
The third digit 8 ways because the second digit cannot be the same as the first and second.
The fourth digit 7 ways for the same reason as the third digit.
Using the product rule = i = 9*9*8*7 = 4536 have distinct digits
(d) Use the quotient rule = i = 9000/3 = 3000
(e) Use the quotient rule = i5 = 9000/5 = 1800
Use the quotient rule = i7 = 9000/7 = 1286
the integers divisible by 5 and 7 is 5*7 = 35
i35 = 9000/35 = 257
Number of integers divisible by 5 or 7 can be determined by the subtraction rule therefore:
i = i5+i7-i35 = 1800+1286-257 = 2829
(f) The integers divisible not divided 5 or 7 is the same as the integers not divisible by 3 or 4:
i(not 5 or 7) = |X| - i(5 or 7) = 9000-2829 = 6171
(g) Integers divisible by 11 but not 7 are the same as integers divisible by 11 but not divisible by 7 and 11.
i(5 not by 7) = i5-i35 = 1800 - 257 = 1543
(h) i(5 and 7) = i35 = 257
This solution to this mathematical problem is resolved using the principle of Integers and basic rules of mathematical operations.
What is an Integer?
An integer in mathematics is simply a whole number. That is a number that is not a fraction.
The following rules govern integers and we shall explore them in the course of the solution:
- Product Rule
- Division Rule
- Subtraction Rule
- Addition Rule
Note that there are 9000 integers between 1000 and 9999.
a) So how many positive integers between 1000 and 9999 inclusive that are divisible by 9?
The total number of integers that exist between 1000 and 9999 inclusive of both figures that can be divided by 9 is gotten using the Division Rule.
The division rule states that:
In a finite set such a A, if the finite set has no overlapping elements, where the elements are represented by d, then
n = | A |/d
Hence, we have | A| = 9000 and d = 1000, therefore, n = 9000/1000 = 9
b) So how many positive integers 1000 and 9999 inclusive are even?
The division rule applies here. An even number is defined as one that is divisible by 2. Hence, 9000/2 = 4,500
c) So how many positive integers between 1000 and 9999 inclusive have distinct digits?
In number numerals, there are 10 distinct digits.
- Because the initial digit cannot be zero, you can split it in nine ways.
- The second digit can be zero, but it can't be the same as the first, therefore there are nine possibilities.
- Because the second digit cannot be the same as the first and second, the third digit is used eight times.
- For the same reason as the third digit, the fourth digit is used seven times.
So we apply the product rule:
9 x 9 x 8 x 7 = 4536
d) are not divisibly by 3?
The product rule also applies here. The total number divisible by 3 is given as:
| 9000| /3 = 3000. Hence those not divisible by 3 =
9000-3000 = 6,000.
e) are divisibly by either 5 or 7?
applying the product rule, we can say
|9000|/5 = 1,800
|9000|/7 = 1285.71428571 ≈ 1286.
But we also have integers divisibly by 5 AND 7 (that is, 5x7). Hence
|9000|/(5x7) ≈ 257
So, the number of Integers divisible by 5 or 7 = (integers divisible by 5 + integers divisible by 7) less (integers divisible by both 5 and 7)
that is,
(1800+1286) - 257
The number of Integers divisibly by 5 or 7 from the selection is 2,829.
(f) are not divisible by either 5 or 7?
The number of integers that are not divisible by either 5 or 7 is given as:
Total Number of integers - Number of Integers divisible by 5 or 7
Recall that the Number of Integers divisible by 5 or 7 from e above is 2829. Hence
= 9000 - 2829 = 6171.
Hence, the number of integers that are not divisible by either 5 or 7 is 6171.
(g) are divisible by 5 but not by 7?
This is given by:
integers divisible by 5 - integers divisible by both 5 and 7. From previous computation, we already have figures for the above. So:
1800-257 = 1,543.
(h) are divisible by 5 and 7?
Recall also that this has been computed using the product rule. Where 5 and 7 is 5 x 7 = 35.
Hence
|9000|/35 = 257.
Therefore, there are 257 integers that are divisible by 5 and 7.
Learn more about Integers at:
https://brainly.com/question/24653557
