2 years ago

A3-inch candle burns down in 5 hours. How far has it burned after 8

1 answer:

5 0

Answer:Assuming a 3-inch diameter, pillar candles of 3, 4 and 6 inches in height will burn for 45, 60 and 75 hours, respectively. Three-inch Party Lite pillar candles will burn 30-35 hours; 5-inch, 65 to 70 hours; and 7-inch, 90 to 95 hours. The three-wick 6-by-5-inch candles burn for more than 100 hours.

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Given $a \equiv 1 \pmod{7}$, $b \equiv 2 \pmod{7}$, and $c \equiv 6 \pmod{7}$, what is the remainder when $a^{81} b^{91} c^{27}$

Blizzard [7]

$\begin{cases}a\equiv1\pmod7\\b\equiv2\pmod7\\c\equiv6\pmod7\end{cases}$

$a^{81}\equiv1^{81}\equiv1\pmod7$

$b^{91}\equiv2^{91}\pmod7$

$2^{91}\equiv(2^3)^{30}\times2^1\equiv8^{30}\times2\pmod7$
$8\equiv1\pmod7$
$2\equiv2\pmod7$
$\implies2^{91}\equiv1^{30}\times2\equiv2\pmod7$

$c^{27}\equiv6^{27}\pmod7$

$6^{27}\equiv(-1)^{27}\equiv-1\equiv6\pmod7$

$\implies a^{81}b^{91}c^{27}\equiv1\times2\times6\equiv12\equiv5\pmod7$