Math Problem Statement
What will be the reminder when 2^100 is divided by 3 3^1989 is divided by 7 128^1000 is divided by 1000
Solution
Let's calculate the remainders for each expression:
1. divided by 3
To find the remainder when is divided by 3, we can use modular arithmetic. First, note the pattern in the powers of 2 modulo 3:
2^1 &\equiv 2 \pmod{3}, \\ 2^2 &\equiv 4 \equiv 1 \pmod{3}, \\ 2^3 &\equiv 2 \times 1 \equiv 2 \pmod{3}, \\ 2^4 &\equiv 2 \times 2 \equiv 4 \equiv 1 \pmod{3}. \end{aligned}$$ The pattern repeats every 2 powers: $$2, 1, 2, 1, \dots$$ Since 100 is even, $$2^{100} \equiv 1 \pmod{3}$$. **Remainder**: $$1$$. ### 2. $$3^{1989}$$ divided by 7 Next, consider $$3^{1989}$$ modulo 7. Again, observe the pattern in the powers of 3 modulo 7: $$\begin{aligned} 3^1 &\equiv 3 \pmod{7}, \\ 3^2 &\equiv 9 \equiv 2 \pmod{7}, \\ 3^3 &\equiv 3 \times 2 \equiv 6 \pmod{7}, \\ 3^4 &\equiv 3 \times 6 \equiv 18 \equiv 4 \pmod{7}, \\ 3^5 &\equiv 3 \times 4 \equiv 12 \equiv 5 \pmod{7}, \\ 3^6 &\equiv 3 \times 5 \equiv 15 \equiv 1 \pmod{7}. \end{aligned}$$ The pattern repeats every 6 powers: $$3, 2, 6, 4, 5, 1, \dots$$ To find $$3^{1989} \mod 7$$, we first compute $$1989 \mod 6$$: $$1989 \div 6 = 331 \text{ remainder } 3$$ So, $$3^{1989} \equiv 3^3 \equiv 6 \pmod{7}$$. **Remainder**: $$6$$. ### 3. $$128^{1000}$$ divided by 1000 For this, we can use **Euler's theorem** or directly compute using the property of powers. First, observe that $$128 \equiv 128 \pmod{1000}$$. Next, we calculate $$128^2 \equiv 16384 \equiv 384 \pmod{1000}$$. Calculating more powers modulo 1000: $$128^4 \equiv 384^2 \equiv 147456 \equiv 456 \pmod{1000}$$ $$128^8 \equiv 456^2 \equiv 207936 \equiv 936 \pmod{1000}$$ $$128^{16} \equiv 936^2 \equiv 876096 \equiv 96 \pmod{1000}$$ After a few more calculations or by using a fast exponentiation method, it is established: $$128^{1000} \equiv 0 \pmod{1000}$$ This is because any power of a number divisible by 2 (with a base that’s a multiple of 128) raised to a large enough power andAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Exponentiation
Formulas
Euler's theorem
Theorems
Modular exponentiation properties
Suitable Grade Level
Advanced High School
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