Question

how many positive integers between 1000 and 9999 inclusive (a) are divisible by 9? (b) have distinct digits? (c) are divisible by 5 or 7? (d) are divisible by 5, but not 7?

Answer 1

(a) The number of positive integers divisible by 9 is 1000.

(b) The number of distinct digits is 4536.

(c) The numbers divisible by 5 or 7 are 2829.

(d) The numbers divisible by 5 but not 7 is 1543.

In mathematics, a whole number is known as an integer. This particular number is not a fraction.

In the course of the solution, we will examine the rules that apply to integers.

The rule for Products: Addition Rule, Subtraction Rule, Division Rule

The number of integers between 1000 and 9999 is 9,001.

(a) Using the Division Rule, one may obtain the entire number of integers between 1000 and 9999, inclusive of both numbers that can be divided by 9.

According to the division rule, given a finite set like A, if the finite set has no overlapping members and the elements are represented by d.

Then,

n = |A| / d

Therefore, n = 9000 / 9 = 1000

(b) There are 10 unique digits in numbers.

• You have nine possible methods to split the first digit because it cannot be zero.
• There are nine options because the second digit might be zero but cannot be the same as the first.
• The third digit is used eight times since the second digit cannot be the same as the first and second.
• The fourth digit is utilized seven times for the identical purpose as the third digit.

Therefore, by product rule:

= 9 × 9 × 8 × 7 = 4536

(c) By product rule,

| 9000 | / 5 = 1800

| 9000 | / 7 = 1286

But there are also integers that can be divided by 5 AND 7. (that is, 5x7).

Therefore,

| 9000 | / ( 5 x 7 ) = 257

In other words, the amount of integers that are divisible by either 5 or 7 is equal to the sum of integers divisible by 5 and 7 less the integers divisible by both 5 and 7

So,

= ( 1800 + 1286 ) - 257

= 2829

(d) The numbers divisible by 5 but not 7 will be:

= integers divisible by 5 - integers divisible by both 5 and 7

= 1800 - 257 = 1543

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