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Question 1165100: (a) Find the smallest positive integer which can be written as the sum of
three, four, five or seven consecutive integers.
(b) Explain why it is not possible for a number to be written as both
the sum of four consecutive integers and the sum of six consecutive
integers.
Found 2 solutions by math_helper, greenestamps:
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! "Integers" includes negative values, unless otherwise restricted.
-1 + 0 + 1 = 0
0 + 1 + 2 = 3 (3 is the smallest positive integer that is the sum of three consecutive integers)
-1 + 0 + 1 + 2 = 2 (2 is the smallest positive integer that is the sum of four consecutive integers)
etc.
5 consec. integers: min sum gt zero = 5
7 consec. integers: min sum gt zero = 7
(b) if N = a + (a+1) + (a+2) + (a+3) = 4a + 6
and N = b + (b+1) + (b+2) + ... + (b+5) = 6b + 15
then 6b + 15 = 4a + 6
6b + 9 = 4a
and a = (6b + 9) / 4
which is not an integer (6b is even, even + odd is odd, odd divided by 4 has a remainder)
To the student: Oh, I may have misread the question. I see what you mean... perhaps the question should have been "Find the smallest positive integer which can be written as the sum of three, four, five, AND seven consecutive numbers." but OR makes sense too upon re-reading.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
(1) The sum of n consecutive integers is n, multiplied by the average of the first and last terms.
(2) If n is odd, then the first and last terms are either both even or both odd; that means the sum of the first and last terms is even, so the average of the first and last terms is an integer.
(3) If n is even then the first and last terms are one even and one odd; that means the sum of the first and last terms is odd, so the average of the first and last terms is halfway between two integers.
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(a) Find the smallest positive integer which can be written as the sum of three, four, five or seven consecutive integers.
Item (2) above tells us that the sum of 3 consecutive integers is divisible by 3, the sum of 5 consecutive integers is divisible by 5, and the sum of 7 consecutive integers is divisible by 7. So a number that can be written as a sum of 3 or 5 or 7 consecutive integers has to be divisible by 3 and by 5 and by 7; that means it has to be divisible by the LCM of 3, 5, and 7, which is 3*5*7 = 105.
So 105, and any positive integer multiple of 105, can be written as the sum of 3, 5, or 7 consecutive integers.
So the positive integers that can be written as the sum of 3, 5, or 7 consecutive integers are of the form 105k, where k is a positive integer.
We are looking for the smallest positive integer that can be written as the sum of 3, or 4, or 5, or 7 consecutive integers. We know that any number of the form 105k can be written as the sum of 3 or 5 or 7 consecutive integers; what additional requirement do we need to make a number of that form also able to be written as the sum of 4 consecutive integers?
Item (3) above tells us that the average of n consecutive integers is halfway between two integers if n is even.
If the number 105k is to be written as the sum of 4 consecutive integers, then item (3) above tells us that 105k divided by 4 must be halfway between two integers. That is going to be true only if k is even.
So the positive integers that can be written as the sum of 3, or 4, or 5, or 7 consecutive integers are of the form 105(2k), or 210k.
Then of course the smallest of those is 210.
ANSWER, part (a): 210
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(b) Explain why it is not possible for a number to be written as both the sum of four consecutive integers and the sum of six consecutive integers.
Item (3) above tells us that the sum of 4 consecutive integers is 4 times a number halfway between two integers: 4(a+1/2) = 4a+2. That sum is always even.
Item (3) above also tells us that the sum of 6 consecutive integers is 6 times a number halfway between two integers: 6(b+1/2) = 6b+3. That sum is always odd.
So the sum of 4 consecutive integers is always even, while the sum of 6 consecutive integers is always odd. So clearly there are no numbers that can be written both as the sum of 4 consecutive integers and as the sum of 6 consecutive integers.
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