Using inclusion-exclusion: patients with improvement or side effects = 100 + 120 – 36 = <<100+120-36=184>>184.
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- Patients with improvement: (1/3) × 300 = <<300/3=100>>100.
- Patients with side effects: 40% of 300 = 0.4 × 300 = <<0.4*300=120>>120.
- Patients with both: 12% of 300 = 0.12 × 300 = <<0.12*300=36>>36.
- But we want improvement only: 100 – 36 = <<100-36=64>>64.
- A wildlife AI model predicts that 60% of a 250 elk herd will migrate north, while 20% stay behind. Of those migrating, 15% deviate from the predicted route. How many migrating elk stay on the correct path?
- Number expected to migrate: 60% of 250 = 0.6 × 250 = <<0.6*250=150>>150.
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At least one of the four is divisible by $ 3 $
Now consider $ 15 $. Since $ 3 $ and $ 5 $ always divide the product, the product is divisible by $ 15 $.
But we can go further: among four consecutive odd numbers, at least one is divisible by $ 3 $, and at least one divisible by $ 5 $. Also, at least two of them are congruent modulo $ 4 $, so their product includes multiple factors of $ 2 $, but since all are odd, no power of $ 2 $ is introduced.
So the largest integer that must divide the product of any four consecutive odd integers is $ 105 $.
Solution: Let the number be $ x $. We are given:
Let $ x = 7k + 1 $. Substitute into second congruence:
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