Let A be the largest positive integer that divides all the numbers of the form 3k + 4k + 5k and B be the largest positive integer that divides all the numbesr of the form 4k + 3(4k) + 4k+2, where k is any positive integer. Then (A + B) equals
Explanation:
Given, 3k + 4k + 5k put k = 1, we have 3k + 4k + 5k = 3 + 4 + 5 = 12 put k = 2, we have 3k + 4k + 5k = 9 + 16 + 25 = 50
The only common factor between 12 and 50 is 2.
3k + 4k + 5k is always even for any value of k, and hence always divisible by 2.
∴ A = 2
Given, 4k + 3(4k) + 4k+2 = 4k(1 + 3 + 42) = 4k × 20 For k ≥ 1, this number is always divisible by 4 × 20 = 80 ∴ B = 80
⇒ A + B = 2 + 80 = 82.
Hence, 82.
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