Improve your reasoning power and Solve the Mathematics equation problem

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)