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Anton [14]
2 years ago
5

WHAT IS THE REMAINDER WHEN

alt="32^{37^{32} }" align="absmiddle" class="latex-formula"> IS DIVIDED BY 9?
Mathematics
1 answer:
Feliz [49]2 years ago
7 0

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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find the average rate of change from t=2 to t=5 for the velocity function v(t)=t^2-t+10. WILL GIVE THE BRAINLIEST ANSWER--EXPLAI
Elina [12.6K]
Answer: The average rate of change is 6.
First, plug in each value of <em>t</em> into the function, v(t) to find there coordinate pairs.

v(2) = (2)^2 - (2) + 10
v(2) = 4 + 8
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v(5) = (5)^2 - (5) + 10
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You can write these values as coordinate pairs, like so: (2, 12) and (5, 30).
The formula for the average rate of change is A(x) =  \frac{f(x)-(f(a)}{x-a}. When you plug in the values from this particular case, the average rate of change formula becomes A(t) = \frac{30-12}{5-2}, or A(t)= \frac{18}{3}.

Looking at the equation, you can solve for the average rate of change between t = 2 and t = 5, which equals 6.
8 0
3 years ago
16. David wanted to buy a new suit selling for $400. He negotiated with the salesperson
Reil [10]
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3 years ago
M= i + r ; solve for r
Alex777 [14]

Answer:

m - i = r

Step-by-step explanation:

In this question, you are solving for "r".

Solve:

M= i + r

We have to get "r" by itself, so subtract "i" from both sides.

m - i = r

Since we found what "r" is, we can't solve any further.

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