1970 AHSME Problems/Problem 4
Problem
Let be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that
Solution
Consider consecutive integers and . Exactly one of these integers must be divisible by 3; WLOG, suppose is divisible by 3. Then and . Squaring, we have that and , so . Therefore, no member of is divisible by 3.
Now consider more consecutive integers and , which we will consider mod 11. We will assign such that and . Some experimentation shows that when so . Similarly, so , and so . Therefore, , so there is at least one member of which is divisible by 11. Thus, is correct.
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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