## The sum of all integers between 500 and 2500 that are divisi

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### The sum of all integers between 500 and 2500 that are divisi

by [email protected] » Thu Mar 28, 2019 5:51 am

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GMATH practice exercise (Quant Class 16)

The sum of all integers between 500 and 2500 that are divisible by both 18 and 75 is:

(A) 6750
(B) 6300
(C) 5400
(D) 4050
(E) 3150

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by [email protected] » Thu Mar 28, 2019 11:06 am
[email protected] wrote:GMATH practice exercise (Quant Class 16)

The sum of all integers between 500 and 2500 that are divisible by both 18 and 75 is:

(A) 6750
(B) 6300
(C) 5400
(D) 4050
(E) 3150
$$? = \sum N \,\,\,::\,\,\,\left\{ \matrix{ \,500 < N\,\,{\mathop{\rm int}} < 2500\,\,\,\,\left( * \right) \hfill \cr \,{N \over {2 \cdot {3^2}}} = {\mathop{\rm int}} \,\,\,;\,\,\,{N \over {3 \cdot {5^2}}} = {\mathop{\rm int}} \,\,\,\,\left( {**} \right) \hfill \cr} \right.$$
$$\left( {**} \right)\,\,\,\, \Rightarrow \,\,\,\,N = k \cdot LCM\left( {2 \cdot {3^2};3 \cdot {5^2}} \right) = k \cdot 2 \cdot {3^2} \cdot {5^2} = 450 \cdot k\,\,\,\,\,\left( {k\,\,{\mathop{\rm int}} } \right)$$
$$\left\{ \matrix{ \,450 \cdot 2 = 900 \hfill \cr \,\left( {450 \cdot 2} \right) \cdot 3 = 2700 \hfill \cr} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,k = 2,3,4,5$$
$$? = 450\,\left( {2 + 3 + 4 + 5} \right)\,\,\, = \,\,\,6300$$