Number System - Questions & Discussions

MBA CAT 2022
624

2. 2^5000 % 525


Euler(525) = 240
2^5000 mod 525 = 2^200 + (240*20) mod 525 = 2^200 mod 525
2^200 mod 525 = 2^200 mod 5^2 * 3 * 7
2^200 mod 3 = 1
2^200 mod 7 = 2
2^200 mod 25 = 1

3a+1 = 7b+2 = 25c+1
75k + 1 = 7b+2
b = (75k - 1)/7
k = 3
Remainder = 226
625

3. 3^1000 % 286


3^1000 mod 286
Euler(286) = 120
3^40 mod 286
Remainder = 133
626

4. 4^2002 % 924


4^2002 = 2^4004 mod 924
2^4002 mod 231
2^4002 mod 7 = 1
2^4002 mod 11 = 4
2^4002 mod 13 = 12
7a+1 = 11b+4 = 13c+12
77k+92 = 13c+12
Remainder = 246
246*4 mod 924 = 60
Remainder = 60
627

A & B take certain decimal number and expresses it in base 5 & 6 respectively. Then A comes and he takes the two representations & assuming that the expressions are in base 10, adds the numbers. Which of the following can't be the unit's digit of sum obtained.
1)0 2)2 3) 8 4)6 5)3

Any help would be much appreciated. Thanks!

i have a problem solving questions with different bases. Can anyone give me right approach to tackle this kind of problems?

628
A & B take certain decimal number and expresses it in base 5 & 6 respectively. Then A comes and he takes the two representations & assuming that the expressions are in base 10, adds the numbers. Which of the following can't be the unit's digit of sum obtained.
1)0 2)2 3) 8 4)6 5)3

Any help would be much appreciated. Thanks!

i have a problem solving questions with different bases. Can anyone give me right approach to tackle this kind of problems?



it should be 8 as 4 + 4 is not possible simultaneously with base 4 and 5

hence 3) 8
629

naaaaaaaaaaa.. anwer is d) 6...
someone plz expalin

630
A & B take certain decimal number and expresses it in base 5 & 6 respectively. Then A comes and he takes the two representations & assuming that the expressions are in base 10, adds the numbers. Which of the following can't be the unit's digit of sum obtained.
1)0 2)2 3) 8 4)6 5)3

Any help would be much appreciated. Thanks!

i have a problem solving questions with different bases. Can anyone give me right approach to tackle this kind of problems?


I think there is somthing wrong with the ques.........
As all the values are possible..........
For eg ur ans 6....could be contradicted by say 22........
22 in base 5 = 42
22 in base 6 = 34

Adding these two no. will result in unit place 6..........

As a no. in base 5 will have their unit digit as ...........
0,1,2,3,4
and no. in base 6 will have their unit digit as...........
0,1,2,3,4,5

So by these all the options are possible........

Check the question again........or the options.....
631

This is a questions from carer launcher.. Here is the solution they have given. I couldnt get it.. can someone plz explain this.

Unit's digits in n2 is one of {0, 1, 4 , 5, 6 , 9}
That means, in base 5, units digit possible is one of {0, 1 , 4} only.
Now, note that any natural number is of the form either 3n - 1 or 3n or 3n + 1
Square of any natural number is of the form
9n2 6n + 1 or 9n2
In base 6, units digit could be one of {0, 1, 3, 4} only. So, unit's digit of the sum can never be 6 or 9.

632

Sorry i missed the "square" word. :P

The correct question reads as : A & B take the square of certain decimal number and expresses it in base 5 & 6 respectively. Then A comes and he takes the two representations & assuming that the expressions are in base 10, adds the numbers. Which of the following can't be the unit's digit of sum obtained.
1)0 2)2 3) 8 4)6 5)3

633

Here goes another question:

The product of two numbers '231' and 'ABA' is 'BA4AA' in a certain base system (where base is less than 10), where A and B are distinct digits. What is the base of that system?

1)5 2)6 3)7 4)8 5)4

634
This is a questions from carer launcher.. Here is the solution they have given. I couldnt get it.. can someone plz explain this.

Unit's digits in n2 is one of {0, 1, 4 , 5, 6 , 9}
That means, in base 5, units digit possible is one of {0, 1 , 4} only.
Now, note that any natural number is of the form either 3n - 1 or 3n or 3n + 1
Square of any natural number is of the form
9n2 6n + 1 or 9n2
In base 6, units digit could be one of {0, 1, 3, 4} only. So, unit's digit of the sum can never be 6 or 9.


It is talking about the squares hence in the ques it must be specified that "the square of those no. are added then which of the following could not be the unit digit of the summation"..........
635

Thanks Shashank! please see my last post. my apologies for missing out a word..
Anways i have posted another question! try solving that 😃

636
637
Here goes another question:

The product of two numbers 231 and ABA is BA4AA in a certain base system (where base is less than 10), where A and B are distinct digits. What is the base of that system?

1)5 2)6 3)7 4)8 5)4


Answer is 6 again.............
638

yeppie.. correct answer.. could you please explain your logic for this ?

639
640
Let base = n then.....

(2n^2 + 3n + 1)(An^2 + Bn + A) = BA4AA

(2n+1)(n+1)(An^2 + Bn + A) = BA4AA

that means BA4AA is divisible by n+1, therefore
(A+4+B) - (A+A) = 0 or (n+1)*x,where x is an integer.

let x = 1,
4 + B - A = n + 1 =>
n = B - A + 3
Compare again we will have
A = 1, B = 4
so, n = 6

Compare again with wat???
641
jain_ashu Says
Compare again with wat???


By compare again I meant follow the same, as (2n+1) is also a factor. But that time I did calculation mistake and found ans as 6 which was though correct........

I am not getting it again...........

That means it can't be done this way........
642

each coefficient in the equation ax^2+bx+c is determined by throwing an ordinary dice(six faced).find the probability that the eqn will have real roots?
plz ans wid full appraoch

643
Here goes another question:

The product of two numbers '231' and 'ABA' is 'BA4AA' in a certain base system (where base is less than 10), where A and B are distinct digits. What is the base of that system?

1)5 2)6 3)7 4)8 5)4

jain_ashu Says
Compare again with wat???

By compare again I meant follow the same, as (2n+1) is also a factor. But that time I did calculation mistake and found ans as 6 which was though correct........

I am not getting it again...........

That means it can't be done this way........


here is link for solution


http://www.pagalguy.com/forum/quantitative-questions-and-answers/42674-official-quant-thread-cat-09-a-774.html#post1655185
For, a digit at tens place of product y + 3x = Base + x (or) 2x + y = Base ---------(1)
For, a digit at hundredth place, 3x+3y = Base + Base + 4 + 1 (carry over)
2y + x + 1 = Base + 4 --------(2)
From (1) and (2),
2y + x + 1 = 2x + y + 4
Y = x +3
For thousandth place of product:
(carry over) 2+3x+2y = Base + Base + x
2+3x+2y = 2 (2x+y) +x
X = 1 and Y = 4
Base = 2 (x) + y = 6
 hope am clear

ps : put x as A and y as B .. jus change of variable