Question 1029985
*[tex \large 51 \equiv 2 \pmod{7}] so *[tex \large 51^{61} \equiv 2^{61} \pmod{7}]


Note that *[tex \large 2^3 \equiv 1 \pmod{7}], so *[tex \large 2^{60} \equiv 1 \pmod{7}]. Then *[tex \large 2^{61} \equiv 2^{60} \cdot 2^1 \equiv 1 \times 2 \equiv 2 \pmod{7}] so the remainder is 2.