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Integers

by adthedaddy » Thu Sep 24, 2015 7:14 pm
What is the smallest positive integer that is divisible by every odd integer between 1 and 10 inclusive?

A) 105
B) 315
C) 945
D) 3780
E) 7560

Please help
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by GMATGuruNY » Thu Sep 24, 2015 7:51 pm
adthedaddy wrote:What is the smallest positive integer that is divisible by every odd integer between 1 and 10 inclusive?

A) 105
B) 315
C) 945
D) 3780
E) 7560
Let x = the smallest positive integer divisible by the odd integers between 1 and 10, inclusive.
Thus, x must divisible by all of the following:
1, 3, 5, 7, 9.

Since 5*7*9 = 315, x cannot be less than 315.
Eliminate A.

Like all positive integers, 315 is divisible by 1.
Since 315 is divisible by 9, it must also be divisible by 3.
Thus, x=315.

The correct answer is B.
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by Matt@VeritasPrep » Fri Sep 25, 2015 12:14 am
The question is asking for the LCM of 3, 5, 7, and 9. 3 is a factor of 9, so we can simply find the LCM of 5, 7, and 9.

The numbers share no prime factors, so the LCM is simply 5 * 7 * 9, or 315.
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by [email protected] » Fri Sep 25, 2015 9:18 am
Hi adthedaddy,

This prompt can be dealt with in a couple of different ways. Even if you don't spot the 'concept' that this question is based on, you can still get to the solution with a bit of basic arithmetic. Since the answers to the question are all numbers, and the prompt asks for the SMALLEST number that is divisible by 1, 3, 5, 7 and 9... we can TEST THE ANSWERS.

Let's start with Answer A: 105

If you know the 'rule of 9', then you know that 105 is NOT evenly divisible by 9. Even if you don't know that rule though, a bit of quick division will prove that 105 is not divisible by 9.

105/9 = 11 2/3
Eliminate Answer A.

Next, Answer B: 315

315 IS divisibly by 9 (so it's also divisible by 3), it's divisible by 5 (since it ends in a 5) and it's also divisible by 7.

315/9 = 35
315/5 = 63
315/7 = 45
So this MUST be the answer.

Final Answer: B

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