Modulo MCQ Quiz in मराठी - Objective Question with Answer for Modulo - मोफत PDF डाउनलोड करा

Concept:

(X + Y) mod M = X mod M + Y mod M

Calculation:

X = 48, Y = 15 and X mod Y = (X + kY) mod Y

∴ X mod Y = 48 mod 15 = 3

(48 + k 15) mod 15 = 48 mod 15 + 15k mod 15

⇒ (48 + k15) mod 15 = 3 + 0 = 3 for all integers k

Thus k can be any integer.

Therefore option 4 is correct.

Concept:

Totient function: 

For n ≥ 1, the totient function denoted by ϕ(n) is the number of positive integers not exceeding n ( ≤ n ) and relatively prime to n.

Euler´s theorem :

If n ∈ Z⁺ and gcd(a, n) = 1 then, a^{ϕ (n)} ≡ 1(mod n)   

If a ≡  b (mod n) and  c ≡  d (mod n) then a + c ≡ b + d (mod n) where a, b, c, d are any integers       ....(5)

Calculation:

Given m and n are relatively prime 

i.e., gcd(m, n) = 1

Then we have ,

mϕ (n) ≡ 1(mod n)       ....(1)

and nϕ (m) ≡ 1(mod m)       .....(2)

Also,

mϕ (n) ≡ 0(mod n)       ....(3)

and nϕ (m) ≡ 0(mod m)       .....(4)

Using (5), we have

m^{ϕ (n)} + n^{ϕ (m)} ≡ 1 + 0 (mod n)

⇒ mϕ (n) + nϕ (m) ≡ 1 (mod n)

And since gcd(m, n) = 1

we have,

mϕ (n) + nϕ (m) ≡ 1 (mod mn)

Hence,  the correct answer is option 3)