How many of the whole numbers from 1 to 100 are either divisible by 7 or include a 7 in their digits?

To solve this numerical puzzle, we must first identify two distinct groups of numbers from 1 to 100. The first group includes all numbers that are multiples of seven. There are 14 such numbers, beginning with 7 itself and ending with 98. The second group is composed of any number that has the digit "7" as part of it. This set includes numbers such as 17, 27, and the entire block of numbers from 70 to 79, which totals 19 numbers.

A simple addition of these two sets of numbers would lead to an inaccurate total because some numbers qualify for both categories. For instance, the numbers 7, 70, and 77 are all divisible by seven and also contain the digit seven. To find the correct count, we must add the numbers from our two initial groups together (14 + 19) and then subtract the three numbers that overlap. This ensures they are not counted twice and gives us the final correct total of 30 numbers.

This common counting issue is a simple illustration of a mathematical idea known as the principle of inclusion-exclusion. Beyond mathematics, the number seven is significant in many cultures, often associated with luck and występujący in contexts like the seven days of the week or the seven wonders of the ancient world. As a prime number, it possesses