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Question Number 99584 by DGmichael last updated on 21/Jun/20

Answered by Rasheed.Sindhi last updated on 22/Jun/20

If a≡b(mod m) and P(x) is a polynomial with integer coficients then P(a)≡P(b)(mod m) Now, Let P(x)=a_n x^n +a_(n−1) x^(n−1) +...+a_1 x+a_0 where 0≤a_i ≤9 As 10≡1(mod 9) Therefore, a_n .10^n +a_(n−1) .10^(n−1) +...a_1 .10+a_0 ≡a_n .1^n +a_(n−1) .1^(n−1) +...+a_1 .1+a_0 (mod 9) a_n a_(n−1) ...a_1 a_0 ≡a_n +a_(n−1) +...+a_1 +a_0 (mod 9) That is a number is divisible by 9 if its sum of digits is divisible by 9.

Ifa≡b(modm)andP(x)isapolynomialwithintegercoficientsthenP(a)≡P(b)(modm)Now,LetP(x)=anxn+an−1xn−1+...+a1x+a0where0⩽ai⩽9As10≡1(mod9)Therefore,an.10n+an−1.10n−1+...a1.10+a0≡an.1n+an−1.1n−1+...+a1.1+a0(mod9)anan−1...a1a0≡an+an−1+...+a1+a0(mod9)Thatisanumberisdivisibleby9ifitssumofdigitsisdivisibleby9.

Commented by DGmichael last updated on 23/Jun/20

thanks sir !��

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