I hope you are familiar with congruences and the chinese remainder theorem.

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The general theorem used here if,

x=a(mod pq) where gcd(p,q)=1

It is equivalent to (Euclid's Lemma),

x=a(mod p) and x=a(mod q)

Thus, by the terms of the problem,

x=9(mod 10)

x=8(mod 9)

x=7(mod 8)

x=6(mod 7)

x=5(mod 6)

Equivalently,

x+1=0(mod 2*5)

x+1=0(mod 3^2)

x+1=0(mod 2^2)

x+1=0(mod 7)

x+1=0(mod 2*3)

Equivalently,

x+1=0(mod 2) (1)

x+1=0(mod 5) (2)

x+1=0(mod 3^2) (3)

x+1=0(mod 2^2) (4)

x+1=0(mod 7) (5)

x+1=0(mod 2) (6)

x+1=0(mod 3) (7)

Equations:

(4) implies (1) and (6)

(3) implies (7)

Thus, eliminating those,

x+1=0(mod 5)

x+1=0(mod 9)

x+1=0(mod 7)

x+1=0(mod 8)

Equivalently,

x=-1 (mod 5)

x=-1 (mod 9)

x=-1 (mod 7)

x=-1 (mod 8)

And, gcd between any two is one.

Thus you can rely on Chinese Remainder Theorem