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Problem of the Week
Problem B and Solution
It All Adds Up!
Problem
What is the least number of consecutive whole numbers that have a sum of
1 000? What are these numbers?
For example, 499 + 500 = 999 , and 500 + 501 = 1001 , so two consecutive
numbers does not work. Do you think three consecutive numbers will work?
Solution
To check whether three consecutive numbers will work, we note that 1000 divided
by 3 is approximately 333. But
332 + 333 + 334 = 999 (too low), and 333 + 334 + 335 = 1002 (too high),
so it appears three consective numbers with a sum of 1000 cannot be found either.
Similarly, if we try four consecutive numbers, we expect terms around 250. But
248 + 249 + 250 + 251 = 998 , and 249 + 250 + 251 + 252 = 1002 ,
so four consective numbers won’t work either.
Finally, since 1000 divided by 5 is 200, we try for five consecutive numbers:
198 + 199 + 200 + 201 + 202 = 1000 .
Thus we see that 5 consecutive numbers will sum to 1000.
Something to Think About
Do you think four consecutive numbers could ever have an odd sum? Why, or
why not?