Challenge for all those who want to BEAT THE GMAT

[rijul007]

Combinatorics

Find the number of all 5 digit numbers(in base 10) each of which contains the block 15 and is divisible by 15. (for eg. 31545, 34155,...)




Remainder

Find the least possible value of (a+b), where a,b are positive integers such that
11 divides a + 13b, and
13 divides a + 11b.


All the best!!

[shankar.ashwin]

Combinatorics

Remainder

Find the least possible value of (a+b), where a,b are positive integers such that
11 divides a + 13b, and
13 divides a + 11b.

a + 13 b = 11K1 -> 13a + 169b = 143K1
a + 11b = 13K2 -> 11a + 121b = 143K2

Subtracting we have ; 2a + 48b = 143(K1-K2) -> a + 24b = 143K

Therefore, a+24b would be a multiple of 143.

Since we want a minimum value of a+b, try to maximize value of 'b' in a+24b = 143. (b=6 gives 144, so 'b' can be max 5)

a + 24(5) = 143 -> a =23.

a+b = 28 IMO.

[rijul007]

Combinatorics

Remainder

Find the least possible value of (a+b), where a,b are positive integers such that
11 divides a + 13b, and
13 divides a + 11b.

a + 13 b = 11K1 -> 13a + 169b = 143K1
a + 11b = 13K2 -> 11a + 121b = 143K2

Subtracting we have ; 2a + 48b = 143(K1-K2) -> a + 24b = 143K

Therefore, a+24b would be a multiple of 143.

Since we want a minimum value of a+b, try to maximize value of 'b' in a+24b = 143. (b=6 gives 144, so 'b' can be max 5)

a + 24(5) = 143 -> a =23.

a+b = 28 IMO.

Good Job Shankar!!

What about the 1st ques..??

[shankar.ashwin]

Honestly found it too complicated and didnt even want to try. Is there a simple/easy way of doing it?

Good Job Shankar!!

What about the 1st ques..??

[Anurag@Gurome]

These problems are way beyond the difficulty level of GMAT. To be precise, they are from previous National (India) level maths Olympiad papers. I'm not going to solve them here as plenty of solutions are already available in the web.

For problem 1 refer to the solution of question number 4.

For problem 2 refer to the solution of question number 2.

[rijul007]

Honestly found it too complicated and didnt even want to try. Is there a simple/easy way of doing it?

Good Job Shankar!!

What about the 1st ques..??

My solution to this problem involves breaking it up into several cases and sub-cases... (7 cases in all)

Even I'm looking for a quicky method to solve this...

Can any of the experts help..??

[rijul007]

These problems are way beyond the difficulty level of GMAT. To be precise, they are from previous National (India) level maths Olympiad papers. I'm not going to solve them here as plenty of solutions are already available in the web.

For problem 1 refer to the solution of question number 4.

For problem 2 refer to the solution of question number 2.

Thanks...
I was mailed these ques from a prep website for CAT(India)

Are these ques even beyond the 700+ level probs in GMAT..?

[Anurag@Gurome]

...Are these ques even beyond the 700+ level probs in GMAT..?

For first question, definitely yes. GMAT is not going to test your patience by giving a such lengthy problem that too with so much details.

The second question is however tests your logical skills. Variant of the same may appear in GMAT.

[user123321]

I agree with Anurag. With the amount of time available to solve a question in GMAT, I dont think they are going to test this type of questions, and also this will give more advantage to people who are from Science background but not for others. The wise decision I might make is skip it unless it is the last problem in the math set.

user123321