Difference between revisions of "1964 AHSME Problems/Problem 28"

(Created page with "== Problem== The sum of <math>n</math> terms of an arithmetic progression is <math>153</math>, and the common difference is <math>2</math>. If the first term is an integer,...")
 
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<math>n(a+n-1) = 153</math>
 
<math>n(a+n-1) = 153</math>
  
Since <math>n</math> is a positive integer, it must be a factor of <math>153</math>.  This means <math>n = 1, 3, 9, 17, 51, 153</math> are the only possibilities.  We are given <math>n>1</math>.
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Since <math>n</math> is a positive integer, it must be a factor of <math>153</math>.  This means <math>n = 1, 3, 9, 17, 51, 153</math> are the only possibilities.  We are given <math>n>1</math>, leaving the other five factors.
  
 
We now must check if <math>a</math> is an integer.  We have <math>a = \frac{153}{n} + 1 - n</math>.  If <math>n</math> is a factor of <math>153</math>, then <math>\frac{153}{n}</math> will be an integer.  Adding <math>1-n</math> wil keep it an integer.
 
We now must check if <math>a</math> is an integer.  We have <math>a = \frac{153}{n} + 1 - n</math>.  If <math>n</math> is a factor of <math>153</math>, then <math>\frac{153}{n}</math> will be an integer.  Adding <math>1-n</math> wil keep it an integer.

Latest revision as of 21:04, 24 July 2019

Problem

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Solution

Let the progression start at $a$, have common difference $2$, and end at $a + 2(n-1)$.


The average term is $\frac{a + (a + 2(n-1))}{2}$, or $a + n - 1$. Since the number of terms is $n$, and the sum of the terms is $153$, we have:

$n(a+n-1) = 153$

Since $n$ is a positive integer, it must be a factor of $153$. This means $n = 1, 3, 9, 17, 51, 153$ are the only possibilities. We are given $n>1$, leaving the other five factors.

We now must check if $a$ is an integer. We have $a = \frac{153}{n} + 1 - n$. If $n$ is a factor of $153$, then $\frac{153}{n}$ will be an integer. Adding $1-n$ wil keep it an integer.

Thus, there are $5$ possible values for $n$, which is answer $\boxed{\textbf{(D)}}$.

See Also

1964 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 27
Followed by
Problem 29
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