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Algebra Level 2

What is the largest positive integer that must be a divisor of the sum of an arithmetic sequence with 90 integer terms?

15 45 10 90

3 solutions

Aaron Tsai
May 24, 2016

Relevant wiki: Arithmetic Progressions

Let $a_1$ be the first term of the sequence and $a_{90}$ be the last term of the arithmetic sequence. Then the sum of the sequence is

$\dfrac{90(a_1+a_{90})}{2}=45(a_1+a_{90})$

Therefore, the largest natural number that must be a divisor of the sequence is $\boxed{45}$ .

Nice problem! Do you think clarifying that the first and last term must be an integer is necessary?

Christopher Boo - 5 years ago

Hmmmm...I guess you're right

Hung Woei Neoh - 5 years ago

Thanks, I have changed the problem accordingly.

Aaron Tsai - 5 years ago

Can not understand why mentioning the first and last term to be integers would be necessary.

45 will still be "the largest positive integer that MUST BE a divisor" of the sum. Hope, you may feel like correcting me if i am wrong..

Ananya Aaniya - 4 years, 11 months ago

The concept of divisors is only applicable to integers: both the divisor and the number that is divided must be an integer. To know more, read the link

Hung Woei Neoh - 4 years, 11 months ago

@Hung Woei Neoh – Yes, I know. But I was trying to know why we can not have divisor of fractions. We do divide fractions with integers.. and further fractions.. no? If we do.. what do we call them? Will read the link. Thanks.

Ananya Aaniya - 4 years, 11 months ago

why on earth (a1 + a90 ) can not be greater than 45??? !!

Ananya Aaniya - 4 years, 11 months ago

.......do you understand what the question is looking for?

Hung Woei Neoh - 4 years, 11 months ago

What is the largest positive integer => (any integer.. >0) that must be a divisor (any integer that divides the unknown sum) of the sum of an arithmetic sequence (any sequence with any 1st term, and any common difference=> any last term => any combo of (a1+a90)=> Still unknown Number=> can actually be infinitely large =>thus can have infinitely large divisor, that can be a positive integer. ) with 90 integer terms ( of course the number of terms has to be integer.. can u find a 2.11th term in a sequence?? the term sequence refers to specific points on the number line.. not specific ranges... )?

=> please correct me where i am wrong in my understanding as stated within the parentheses...

thanks.

Ananya Aaniya - 4 years, 11 months ago

@Ananya Aaniya – Well then, since you already understand the question, we are looking for the largest positive integer that must be a divisor of

$a_1+a_2+a_3+\ldots+a_{90}$

We know that the sum above equals to $45(a_1+a_{90})$ . Now, what must be the largest positive divisor of this? $\boxed{45}$

Yes, there can be larger positive integer divisors for this, but which positive divisor can divide any and all such APs? Definitely $45$ .

Hung Woei Neoh - 4 years, 11 months ago

@Hung Woei Neoh – You have identified it where it irritated me. The question was not supposed to be about "Largest Positive Integer".

The Question was SUPPOSED TO BE about "The Largest Positive Integer In General".

It is not about the "The Largest" . Generality should have asked about for this answer to be correct.

So, This particular question.. HAS NO RIGHT ANSWER at all.

A wrong and Flawed question.... In case u understand the mammoth difference between the presence and absence of the term "any and all" in the question.

In Math.. U are NOT SUPPOSED to take anything for granted... when a specific thing is asked.. the question must be specific aswell. .. Especially at Level 3. (which is showing as the hughest level for this topic.)

Ananya Aaniya - 4 years, 11 months ago

@Ananya Aaniya – Yes, in math you are not supposed to take anything for granted. Take note of the word "must". Like I said, there are larger positive integers which CAN be a divisor of the sum. But it doesn't mean that it MUST be a divisor. What we're looking for is a number that MUST be a divisor of such a sum, regardless of how the progression is like. This is another way of saying "what is the largest positive integer that can divide any sum of an AP of 90 terms"

Hung Woei Neoh - 4 years, 11 months ago

@Hung Woei Neoh – Yes .. got it.

Ananya Aaniya - 4 years, 11 months ago

Oli Hohman
Jun 26, 2016

The sum of a 90 term arithmetic sequence will be equal to 90 a1 + 4005 d, where d is the common difference. The GCF of 90 and 4005 is 45, so you have 45 (2 a1 + 89d). Since this is as far as this can be factored, 45 is the highest divisor of the sum of any 90 term arithmetic series.

Moderator note:

You have to be careful, just because "that is the extent to which it can be factored", doesn't mean that there might not be additional common factors (over all possible values of $a_1$ and $d$ .

For example, in the sequence generated by $n (n+1)(n+2)$ , the GCD of the terms is 6, even though there is no constants that are immediately factorable.

Daniel Sugihantoro
Jul 29, 2016

Get Sum of arithmetic integer 90 : ${ A }_{ n }=an+(n-1)r\\ A_{ 90 }=a+89r\\ { S }_{ 90 }=90a+(1+2+3+...+88+89)r\\ { S }_{ 90 }=90a+(\frac { 1 }{ 2 } \times 89(1+89))\\ { S }_{ 90 }=90a+4005r$

and then for biggest divisor

$\frac { 90a+4005r }{ 45 } =\quad 2a\quad +\quad 89$

and because that the answer is $\boxed{45}$

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