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Minchanka [31]
3 years ago
13

Prove the divisibility of the following numbers:

Mathematics
1 answer:
Brut [27]3 years ago
6 0

Answer:

Step-by-step explanation:

To prove divisibility, we need to factor the divident such that one of its factors matches the divisor.

(I use the notation x|y to denote that x divides y)

(A)

75^{30}|45^{45}\cdot15^{15}\\45^{45}\cdot15^{15}=3^{45}\cdot 15^{45}\cdot 15^{15}=\\=3^{45}\cdot 15^{60}=3^{45}\cdot 15^{30}\cdot 15^{30}=3^{45}\cdot (3\cdot5)^{30}\cdot 15^{30}=\\=3^{45}\cdot 3^{30}\cdot(5\cdot 15)^{30}=3^{45}\cdot 3^{30}\cdot(75)^{30}\\\implies\\75^{30}|3^{45}\cdot 3^{30}\cdot75^{30}

(B)

72^{63}|24^{54}\cdot 54^{24}\cdot2^{10}\\24^{54}\cdot 54^{24}\cdot2^{10}=(2^{162}\cdot 3^{54})\cdot(2^{24}\cdot 3^{72}) \cdot 2^{10}\\=2^{196}\cdot 3^{126}

In this case, it is easier to also factor the divisor to primes:

72^{63}=2^{189}\cdot 3^{126}

Both of these factor must be matched in the dividend in order to prove divisibility, and that indeed turns out to be true:

2^{189}\cdot 3^{126}|2^{196}\cdot 3^{126}\implies\\2^{189}|2^{196}\,\,\mbox{and}\,\,3^{126}|3^{126}

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Find the solution of the given initial value problem. ty' + 2y = sin t, y π 2 = 9, t &gt; 0 y(t) =
Helen [10]

For the ODE

ty'+2y=\sin t

multiply both sides by <em>t</em> so that the left side can be condensed into the derivative of a product:

t^2y'+2ty=t\sin t

\implies(t^2y)'=t\sin t

Integrate both sides with respect to <em>t</em> :

t^2y=\displaystyle\int t\sin t\,\mathrm dt=\sin t-t\cos t+C

Divide both sides by t^2 to solve for <em>y</em> :

y(t)=\dfrac{\sin t}{t^2}-\dfrac{\cos t}t+\dfrac C{t^2}

Now use the initial condition to solve for <em>C</em> :

y\left(\dfrac\pi2\right)=9\implies9=\dfrac{\sin\frac\pi2}{\frac{\pi^2}4}-\dfrac{\cos\frac\pi2}{\frac\pi2}+\dfrac C{\frac{\pi^2}4}

\implies9=\dfrac4{\pi^2}(1+C)

\implies C=\dfrac{9\pi^2}4-1

So the particular solution to the IVP is

y(t)=\dfrac{\sin t}{t^2}-\dfrac{\cos t}t+\dfrac{\frac{9\pi^2}4-1}{t^2}

or

y(t)=\dfrac{4\sin t-4t\cos t+9\pi^2-4}{4t^2}

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andrey2020 [161]

Answer:

y=4/5x+B

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2 years ago
A polygon has an interior angle sum of 2,520°. How many sides does it have?
Alisiya [41]

Answer:

16

Step-by-step explanation:

Remember the formula: (n-2)*180/n

(n-2)*180/n=2520

Just solve!

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