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Question
If 10n + 3.4n+2 + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.
Options
5
3
7
1
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Solution
If 10n + 3.4n+2 + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is 5.
Explanation:
Let P(n) = 10n + 3.4n + 2 + k is divisible by 9, ∀ n ∈ N
P(1) = 101 + 3.41 + 2 + k = 10 + 3.64 + k
= 10 + 192 + k = 202 + k must be divisible by 9.
If (202 + k) is divisible by 9 then k must be equal to 5.
202 + 5 = 207 which is divisible by 9.
= `207/9`
= 23
So, the least positive integral value of k = 5.
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