Binomial Theorem Applications

Question

The remainder when 2¹⁰⁰ is divided by 7 is

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1. **Modulo Search**: We need to find $2^{100} \pmod{7}$. 
   - Note that $2^3 = 8 \equiv 1 \pmod{7}$.
2. **Breakdown Power**: Write $100$ in terms of $3$ ($100 = 3 	imes 33 + 1$).
   - $2^{100} = (2^3)^{33} \cdot 2^1$
3. **Application of Congruence**:
   - $2^{100} \equiv (1)^{33} \cdot 2 \pmod{7}$
   - $2^{100} \equiv 1 \cdot 2 \equiv 2 \pmod{7}$.
4. **Result**: The remainder is **$2$**.

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