Mathematics
Prove that the sum of three consecutive even numbers is divisible by 6.
Mathematics Proofs
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Answer
Let three consecutive even numbers be : 2n, 2n + 2, 2n + 4
Their sum = 2n + 2n + 2 + 2n + 4
⇒ (2n + 2n + 2n) + (2 + 4)
⇒ 6n + 6 = 6(n + 1) = 6k, where k = n + 1.
Clearly 6k is divisible by 6.
Hence, proved that the sum of three consecutive even numbers is divisible by 6.
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You must have had a friend who must have told you to think of a number and do various things to it, and then without knowing your original number, telling you what number you ended up with. Here are two examples. Examine why they work.
(i) Choose a number. Double it. Add nine. Add your original number. Divide by three. Add four. Subtract your original number. Your result is seven.
(ii) Write down any three-digit number (for example, 425). Make a six-digit number by repeating these digits in the same order (425425). Your new number is divisible by 7, 11 and 13.