Here’s a more clever, less tedious answer ...with a solution:
Start the process by assembling multiples of (2) such that the product does not exceed 100.
Do the same with factors of (3). Then assemble the multiples of (2) and (3) .
Count these, add one (1) to the count because (1) doesn’t have a prime greater than (4).
Subtract this count from 100.
\(\left[ {\begin{array}{ccc} 2^1 = 2 \;\;\; | & 3^1 = 3 \; \;\;\; | & 2^1 * 3^1 = 6 \;\;\;\;\;| & 2^1 * 3^2 = 18 \;\;\; | & 2^1 * 3^3 = 54\\ 2^2 = 4 \;\;\; | & 3^2 = 9 \; \;\;\; | & 2^2 * 3^1 = 12 \;\;\;| & 2^2 * 3^2 = 36 \;\;\; | \\ 2^3 = 8 \;\;\; | & 3^3 = 27 \;\;\; | & 2^3 * 3^1 = 24 \;\;\;|& 2^3 * 3^2 = 72 \;\;\;|\\ 2^4 = 16 \;\; | & 3^4 = 81 \;\;\; | & 2^4 * 3^1 = 48\;\;\;|\\ 2^5 = 32 \;\; |& \hspace{4em} | & 2^5 * 3^1 = 96 \;\;\; |\\ 2^6 = 64 \;\; |& \end{array}} \right]\underbrace{_\text {There are nineteen (19) integers ... plus one (1) }} _\text {...don't forget the (1)}\)
\(100 - 20 = 80 \leftarrow \small \text{ The number of positive integers less than or equal to 100 that have (at least) one prime factor that is greater than 4. }\\ \)
GA
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