To find the greatest number of Activities pages that can be created with the given photos, we need to determine the greatest common divisor (GCD) of the number of color photos (35) and the number of black-and-white photos (21). The GCD will tell us the largest number of pages that maintain the same ratio of color to black-and-white photos.
We can start by finding the prime factorization of both numbers:
For 35:
- \(35 = 5 \times 7\)
For 21:
- \(21 = 3 \times 7\)
Next, we identify the common prime factors. The only common prime factor between 35 and 21 is \(7\).
Now we will confirm that \(7\) is indeed the GCD:
- The factors of \(35\) are \(1, 5, 7, 35\).
- The factors of \(21\) are \(1, 3, 7, 21\).
The largest common factor is \(7\).
Thus, the greatest number of Activities pages that can be created is:
\[ \text{Greatest number of pages} = 7 \]
To find the number of color and black-and-white photos per page, we divide the total number of each type of photo by the number of pages:
-
Color photos per page: \[ \frac{35}{7} = 5 \]
-
Black-and-white photos per page: \[ \frac{21}{7} = 3 \]
This means that each page can feature 5 color photos and 3 black-and-white photos while using all available photographs. Therefore, the greatest number of Activities pages the editor can create is
\[ \boxed{7} \]
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