Problem of the Week
Problem B and Solution
Numb Sums
Problem
Many whole numbers can be written as sums of consecutive numbers, i.e.,
numbers which follow one after the other.
For example, 5 = 2 + 3, 24 = 7 + 8 + 9, 30 = 4 + 5 + 6 + 7 + 8, etcetera.
a) Determine all the numbers from 1 to 16 which can
be written as sums of 2 or more consecutive numbers.
b) Which number from 1 to 16 can be written as a
sum of consecutive numbers in the greatest number
of ways?
c) Which number(s) from 1 to 16 cannot be written as
a sum of consecutive numbers? What are the factors
of these numbers?
Solution
a) Numbers which can be written as sums of 2 or more consecutive numbers
are:
1 = 0+1, 3 = 1+2, 5 = 2+3, 6 = 1+2+3, 7 = 3+4, 9 = 2+3+4 = 4+5,
10 = 1+2+3+4, 11 = 5+6, 12 = 3+4+5, 13 = 6+7, 14 = 2+3+4+5,
and 15 = 1 + 2 + 3 + 4 + 5 = 4 + 5 + 6 = 7 + 8 .
b) The number 15 can be written as a sum in three different ways, the greatest
number of ways for the numbers 1 to 16.
c) The numbers 2, 4, 8, and 16 cannot be written as a sum of consecutive
numbers. Their factors are:
2: {1,2}, 4: {1,2,4}, 8: {1,2,4,8}, 16: {1,2,4,8,16}.
Things to Think About
Why is it that all the odd numbers can be written as sums of consecutive
numbers? Is this true of any odd number?
How do the factors of the even numbers which can be written as a sum of
consecutive numbers differ from those of the numbers in part c)?