3 years ago

A store sells a note book for $3 each an does not charge

Mathematics

2 answers:

Andrej [43]3 years ago

8 0

I'm confused whats the question maybe i can help

Law Incorporation [45]3 years ago

6 0

So what your question

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What’s the exponent form of 2*2*5*5

san4es73 [151]

Answer:

The answer is 100

Step-by-step explanation:

2 times 2=4 4 times 5=20 20 times 5/5 times 2=10 then add a zero

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3 years ago

What percent is equivalent to 18/30

FrozenT [24]

18 / 30 =

60 percent

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3 years ago

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A farm produces an average of 61.7 liters of milk per day. About how many days will it take to produce 1,170 liters of milk?

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3 years ago

What is the approximate volume of the sphere? Use 3.14 to approximate pi and round the answer to the nearest hundredth if necess

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3 years ago

WHAT IS THE REMAINDER WHEN <img src="https://tex.z-dn.net/?f=32%5E%7B37%5E%7B32%7D%20%7D" id="TexFormula1" title="32^{37^{32} }"

Feliz [49]

Recall Euler's theorem: if $\gcd(a,n) = 1$ , then

$a^{\phi(n)} \equiv 1 \pmod n$

where $\phi$ is Euler's totient function.

We have $\gcd(9,32) = 1$ - in fact, $\gcd(9,32^k)=1$ for any $k\in\Bbb N$ since $9=3^2$ and $32=2^5$ share no common divisors - as well as $\phi(9) = 6$ .

Now,

$37^{32} = (1 + 36)^{32} \\\\ ~~~~~~~~ = 1 + 36c_1 + 36^2c_2 + 36^3c_3+\cdots+36^{32}c_{32} \\\\ ~~~~~~~~ = 1 + 6 \left(6c_1 + 6^3c_2 + 6^5c_3 + \cdots + 6^{63}c_{32}\right) \\\\ \implies 32^{37^{32}} = 32^{1 + 6(\cdots)} = 32\cdot\left(32^{(\cdots)}\right)^6$

where the $c_i$ are positive integer coefficients from the binomial expansion. By Euler's theorem,

$\left(32^{(\cdots)\right)^6 \equiv 1 \pmod9$

so that

$32^{37^{32}} \equiv 32\cdot1 \equiv \boxed{5} \pmod9$

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1 year ago