__Improve your reasoning power__

**Question**- When a no. is divided by 2, rem. is 1 when divided by 3, rem. is 2. and so on...till it is divided by 10, remainder is 9.What's the number?

**Answer-**

if the no. be n, (n+1) is divisible by all of {2,3,4,..,10}. hence (n+1) is divisible by 2520.

[ 2 |(n+1) & 3 |(n+1)

=> 6 |(n+1) & 4 | (n+1)

=> 12 |(n+1) (12=l.c.m(6,4)) & 5 |(n+1)

=> 60 |(n+1).

continuing in this way 2520 |(n+1)]

therefore any no. of the form n=(2520p-1) will suffice where p is a natural no.

of course the smallest such natural no. is 2519.

__Mathematics equation problem__**X^3-Y^2=2 FIND ALL INTEGRAL SOLUTION OF X,Y**

Answer:

case 1

x-3=y-5

y=x+2

original eq gives

x^3-x^2-4x-2=0

no integral solution for this.

case 2

x-3=y+5

y=x-8

original eq gives

x^3-x^2+16x-66=0

(x-3)(x^2+2x+22)=0

x=3

y=-5

as eq invoves y^2 hence y as wellas -y both satisfies

the pairs are

(3,5)

and

(3,-5)