The Official CAT 2007 Quant Thread

[mohit_ranka]

Originally Posted by jha ji

i am also getting 7 as the ans

As the question asks for the second last digit, not the last digit, the answer is 0.

See my post, few post up, I hope it would be helpful to explain How??

[mohit1984]

Hi
There was a remainder question long time back ...Find rem of 777777^37 divided by 19?
I just wanted to aska doubt:
Actually when 777777/19 then remainder is 12...so cant we apply Euler theorem here in this way 12^18=1 mod19 so 12^36=1 mod19
Hence we will be left with 12/19 only so remainder is 12....But the answer was 7 so obvoisuly I am wrong but i do not know where I am wrong can u help me?
And yes 1 more doubt just show me how can i apply seed method to find remainder for this 777777/19 and for 777777^2/19??I just wanna see how the equation for seed will change between these 2 cases....
I guess for 777777/19 we will have (2^5+2^4+2^3+2^2+2^1+2^0)*7 but i dunno if i am write? And i dunno how this will change when eqn becomes 777777^2/19?
Please clarify my doubts?
And tell me from where did u guys learn this concept to find remainder so that I may also go thru the web page or some book?

[nishish]

People help me in solving this question...

How many natural numbers less than 1000 can be expressed as the difference of two perfect squares in at least one way......

[fultoo_bakar]

very foolish post so deleted....................

[fultoo_bakar]

Originally Posted by nishish

People help me in solving this question...

How many natural numbers less than 1000 can be expressed as the difference of two perfect squares in at least one way......

let us take two numbers to be x and y then

x^2 - y^2 < p where p < 1000. we have to find different values of p.
let us take maximum value of p = 1000
then x^2 - y^2 = 1000
x can take maximum value of 500 because 500^2 - 499^2 = 999, After this value of x the difference will exceed 1000.

now (x+y)(x-y) = p where p < 1000
let us assume
x + y = 2m
x - y = 2n (2m>2n)

then
x = m+n
y = m-n

where both m and n will be intergers. we can infer that both x+y and x - y should be even to give integral values of x and y

now x < 500 and y < 500.....................so no of ways of choosing different m and n where x > y are (250 C 2)/2. hence the answer.............am i right or i have missed something.............

[nishish]

Originally Posted by fultoo_bakar

Is the answer (500 C 2)/2. i just want to confirm the solution and then i will post the method.....alternatively can you post the answer.........
so that i can twist my method to get the ans

Actually i am also not sure about the answer....
but neways as far as I remeber
the options were around
650-750...
So nehow u have to twist 250C2 a bit

[fultoo_bakar]

Originally Posted by fultoo_bakar

let us take two numbers to be x and y then

x^2 - y^2 < p where p < 1000. we have to find different values of p.
let us take maximum value of p = 1000
then x^2 - y^2 = 1000
x can take maximum value of 500 because 500^2 - 499^2 = 999, After this value of x the difference will exceed 1000.

now (x+y)(x-y) = p where p < 1000
let us assume
x + y = 2m
x - y = 2n (2m>2n)

then
x = m+n
y = m-n

where both m and n will be intergers. we can infer that both x+y and x - y should be even to give integral values of x and y

now x < 500 and y < 500.....................so no of ways of choosing different m and n where x > y are (250 C 2)/2. hence the answer.............am i right or i have missed something.............

My answer is exceeding 1000 so it can never be possible.......it will take Lord Krishna's Chkra to twist this answer.................anyways i think my approach is wrong.........will come with better approach soon.............

[hellion]

Sorry, double post

[hellion]

Originally Posted by nishish

People help me in solving this question...

How many natural numbers less than 1000 can be expressed as the difference of two perfect squares in at least one way......

This observation might help

generating a number which is product of a prime number and 2 is not possible
eg 4,6,10

[krsh.vik]

Originally Posted by nishish

People help me in solving this question...

How many natural numbers less than 1000 can be expressed as the difference of two perfect squares in at least one way......

749
all odd numbers and all multiples of 4 less than 100