(a) This is the same as computing 4⁵⁵ (mod 11). We have
4² ≡ 16 ≡ 5 (mod 11)
4³ ≡ 4 • 5 ≡ 20 ≡ 9 (mod 11)
4⁴ ≡ 4 • 9 ≡ 36 ≡ 3 (mod 11)
4⁵ ≡ 4 • 3 ≡ 12 ≡ 1 (mod 11)
Then from here,
4⁵⁵ ≡ (4⁵)¹⁰ • 4⁵ ≡ 1¹⁰ • 1⁵ ≡ 1 (mod 11)
(b) Each term in the sum
4ⁿ + 4ⁿ⁺¹ + 4ⁿ⁺² + 4ⁿ⁺³ + 4ⁿ⁺⁴
has a common factor of 4ⁿ, so this sum is the same as
4ⁿ (1 + 4 + 16 + 64 + 256) = 4ⁿ • 341 = 4ⁿ • 11 • 31
So the sum is indeed divisible by 11 for all integers <em>n</em>.
(c) Since 4ⁿ = (2ⁿ)², we know the sum is also divisible by 11 when <em>a</em> = 2.
<h3>
Answer: </h3><h3>
x = 58</h3><h3>
y = 32</h3>
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Explanation:
If angle x is the reference angle, then the side 9 is the adjacent side (it is the closer leg to angle x). The hypotenuse is 17 as it is opposite the largest angle. The largest side is always opposite the largest angle in a triangle.
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cos(angle) = adjacent/hypotenuse
cos(x) = 9/17
x = arccos(9/17)
x = 58.034281249203
x = 58
Arccosine is the same as inverse cosine, written as
Make sure your calculator is in degree mode.
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Edit:
After finding x, we can find y
x+y = 90
58+y = 90
y = 90-58
y = 32
Answer:
37.7
Step-by-step explanation:
<h2><u>circumference of a circle:</u></h2>
Formula = 2πr
r is the radius
diameter = 2* radius
hence radius = 12/2
radius = 6 yards
2*π*6 = 37.6991
to 3sf
<h3><u>circle circumference is 37.7</u></h3>
Answer:
2/5
Step-by-step explanation:
8/20=2/5
The product is less than either of the factors. Multiplying a number < 1 with 2/3 another number < 1, will give a fraction of a fraction, and part of a fraction is an even smaller fraction. Another way of looking at it: Fractions less than one have numerators smaller than their denominators. When multiplying fractions, you multiply across, so multiplying two fractions that are both less than one will always give a numerator much smaller than the denominator, and a larger denominator makes for a smaller number. I hope this helps, but if you have questions, ask.