SOLUTION: What is the smallest positive integer that can be expressed as the sum of nine consecutive positive integers, the sum of ten consecutive positive integers, and the sum of eleven co

Question 103322: What is the smallest positive integer that can be expressed as the sum of nine consecutive positive integers, the sum of ten consecutive positive integers, and the sum of eleven consecutive positive integers? Explain how you arrived at this number.
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
Before we start, let's be clear.
Positive integers are 1,2,3,4,5,.... and do not include zero. Consecutive means one right after the other (3 and 4 are consecutive integers).
What is the smallest sum from one positive integer?
1
What is the smallest sum from two consecutive positive integers?
1+2=3 Sum from 1 to 2.
What is the smallest sum from three consecutive positive integers?
1+2+3=6 Sum from 1 to 3.
There's a pattern here.
...
What is the smallest sum from ten consecutive positive integers?
The sum from 1 to 10.
1+2+3+4+5+6+7+8+9+10=55
What is the smallest sum from eleven consecutive positive integers?
The sum from 1 to 11.
Or the sum from 1 to 10 plus 11.
1+2+3+4+5+6+7+8+9+10+11=55+11=66
What is the smallest sum from twelve consecutive integers?
The sum from 1 to 12.
Or the sum from 1 to 11 plus 12.
1+2+3+4+5+6+7+8+9+10+11+12=66+12=78
RELATED QUESTIONS
Show that any positive integer N = 7 modulo 8 can never be expressed as the sum of the... (answered by Sphinx pinastri)

What is the smallest integer that could be part of a set of two or more positive... (answered by Theo)

Show that there is only one set of different positive integers, x,y,z that 1=... (answered by Edwin McCravy)

(a) Find the smallest positive integer which can be written as the sum of three, four,... (answered by math_helper,greenestamps)

A series of 384 consecutive odd integers has a sum that is a perfect fourth power of a... (answered by ikleyn,greenestamps)

A series of 384 consecutive odd integers has a sum that is a perfect fourth power of a... (answered by MathLover1,ikleyn,greenestamps)

Prove that there is a positive integer that can be written as the sum of squares of... (answered by Edwin McCravy)

300 is the sum of 15 consecutive even positive integers. What is the greatest even... (answered by Fombitz)

What are 4 consecutive positive integers whose sum equals 5 times the smallest... (answered by oberobic)