Which number gives:
X mod 10=9
X mod 9=8
X mod 8=7
X mod 7=6
X mod 6=5
X mod 5=4
X mod 4=3
X mod 3=2
X mod 2=1

X º 8 (mod 9)

X º 7 (mod 8)

X º 6 (mod 7)

X º 4 (mod 5)

which you can now solve by Chinese Remainder Theorem, as the moduli are pairwise coprime.

Another approach is to note that the condition given is the same as

X º -1 (mod n)

where n ranges from 2 to 10. Now apply a variation of Chinese Remainder Theorem to get the answer.

Kerwin

Or just note that X + 1 is divisible by 2,3,...,10. So it suffices for X + 1 to be divisible by just 5,7,8,9 (since then X + 1 is automatically divisible by 2,3,4,6). So X + 1 is some multiple of 5x7x8x9.

Consider the number 10! (factorial of 10)
all the number from 1 to 10 divide 10!
Now consider the number (10!-1), This number will give a remainder of (d-1) when divided by d, where d is any number from 2 to 10.