Which of the following integers can be written as both the sum of 5 consecutive odd integers and 7 consecutive odd integers?
Choose one:
A) 49 B) 70 C) 140 D) 215 E) 525
First of all notice that both the sum of 5 consecutive odd integers and the sum of 7 consecutive odd integers must be odd. Eliminate B, and C right away.   Next the sum of 5 consecutive odd integers can be written as: (x-4)+(x-2)+x+(x+2)+(x+4)=5x, so the sum must be odd multiple of 5;   Similarly, the sum of 7 consecutive odd integers can be written as: (y-6)+(y-4)+(y-2)+y+(y+2)+(y+4)+(y+6)=7y, so the sum must be odd multiple of 7;   So, the sum must be odd multiple of 5*7= 
<gadget id="calculator">5*7</gadget>
<output>35</output>
35. Only option E this conditions.   Answer: 525.
<result>E</result>
Which of the following integers can be written as both the sum of 5 consecutive odd integers and 7 consecutive odd integers?