School Olympiads - International Mathematics Olympiad (IMO)

Number Theory, Advanced Algebra & Olympiad Geometry question bank (Mathematical Reasoning & Achievers Section)

Practice 51 free School Olympiads - International Mathematics Olympiad (IMO) Mathematical Reasoning & Achievers Section - Number Theory, Advanced Algebra & Olympiad Geometry questions with answers and step-by-step solutions.

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Q1.

SOF IMO (Number Theory): In 'Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms', what is the remainder when $2^{100}$ is divided by $7$?

(A)

$2$ (Since $2^3 \equiv 1 \pmod 7 \implies (2^3)^{33} \times 2^1 \equiv 1 \times 2 = 2$) (Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms)

(B)

$1$

(C)

$4$

(D)

$0$

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Q2.

Combinatorics & Pigeonhole Principle: In 'Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms', what is the minimum number of socks that must be drawn from a drawer containing 10 blue socks and 10 red socks in the dark to guarantee at least one matching pair?

(A)

3 socks (By Pigeonhole Principle with 2 color categories) (Number Theory: Primes, Divisibility Rules, Modular Arithmetic and Cryptarithms)

(B)

11 socks

(C)

2 socks

(D)

4 socks

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