If a and b are positive integers such that...

[aramak]

If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

a) 10
b) 13
c) 18
d) 26
e) 50

I had a question on a possible shortcut for solving this problem. Since we know that "b" is not a divisor of "a," would it be feasible to identify 10, 18, 26, and 50 as possible values for "b" (since all of these #s contain either 2's or 5's as prime factors, resulting in a terminating decimal) and therefore conclude that "13" must be a divisor of "a"?

[jnelson0612]

If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

a) 10
b) 13
c) 18
d) 26
e) 50

I had a question on a possible shortcut for solving this problem. Since we know that "b" is not a divisor of "a," would it be feasible to identify 10, 18, 26, and 50 as possible values for "b" (since all of these #s contain either 2's or 5's as prime factors, resulting in a terminating decimal) and therefore conclude that "13" must be a divisor of "a"?

Hi Aramak,
Let's first discuss what a terminating decimal is. 2.86 is in fact a terminating decimal; 1/3 = .33333333 and so on is a non-terminating decimal.

In addition, "b" can't be 10 because that would make "a" 28.6 which is not an integer.

Here's how I thought about it: 2.86 could be 286/100, so "a" could be 286. I can quickly eliminate answers A and E. I can quickly try to divide 18 into 286 and find that it does not divide in evenly, so eliminate answer C.

I have B and D left. 13 is a factor of 26, so I will only check 26, and it does in fact go in 11 times.

Okay, can I reduce my values? Yes! I could divide each one by 2, and have 143/50=2.86. Okay, now I can get rid of answer D, 26, and choose B, 13.

[hargadonr]

there is a method in the book G1 pg 129 that says:

when a is divided by positive integer b result is 4.35 which of the following could be the remainder when A is divided by B?

the method drops the leading 4 and examines the decimal as follows:
35/100 = 7/20 = R/D ; 20R=7D therefore R has 7 as a factor.

if you follow that method for this question you get
86/100 = 43/50 = R/D ; 50R=43D thefor R has 43 as a factor. but this doesn't work. (the answer says R is div by 11,13 and 143)

likewise if you use the 'full number' method in the CAT answer (286/100 = 143/50 = R/D ; 143D=50R therefore D has 11 and 13 as factors) for the book example you don't get the right result:
435/100 = 87/20 = R/D ; 87D=20R D , R has 3 and 29 as factors

can someone explain which method is correct or how you choose between them? thx.

[tim]

each of these methods is correct for certain types of problems. you have provided two different types of problems here: one where we are interested in a remainder, and the other where we are interested in factors of the numerator. this is the difference between the problems and thus the solution methods..

[JbhB682]

Just out of curiosity, is 50 a divisor of "b" ?

[Sage Pearce-Higgins]

Yes, it does. How did you deduce that?

[JbhB682]

Hi Sage --

This is the remainder formula given in the strategy guides.

Dividend/Divisor =Quotient + Remainder/Divisor

Example :
If A = 8 and B = 5, dividing A by B, you get

8/5 = 1 +(3/5)

Based on the above formula at-least : "5" is the divisor of the "8" based on the terminology of the above formula (even though 5 is not a "factor" of 8)

Hence i thought in this problem too == "b" [what ever the value of b is] is a divisor of "a"

I agree that 13 (OA answer) MUST be a factor of "A" , but frankly the Divisor of A == can be ANY value you choose to divide A by based on the terminology of the above formula in blue

In the GMAT exam -- do you think the word "Divisor" of A would be perhaps replaced with the word "Factor" of A so that there is no confusion

Please let me know if my thinking is accurate or perhaps incorrect

Thank you !

[Sage Pearce-Higgins]

Interesting point of terminology. Although I see that my colleague used the word 'divisor' in this context to mean 'the number on the bottom of the division', my understanding is that in most cases, divisor means the same as factor. I.e. I would say that 5 is not a divisor of 8. There are a couple more OG problems that use the word 'divisor' - check them out for clarification.