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blondinia [14]
2 years ago
5

How many positive integers less than 1000 are 6 times the sum of their digits ?.

Mathematics
1 answer:
Grace [21]2 years ago
4 0

Only 1, 54.

There are obviously no such 1-digit numbers (1 - 9).

Let n = ab be a 2-digit number whose value is 6 times its digit sum, meaning

n = 10a + b = 6 (a + b)

Then

10a + b = 6a + 6b

4a = 5b

Now, 4 and 5 are coprime, so for equality to hold, a must be a multiple of 5 and b must be a multiple of 4.

The only valid multiple of 5 is 5 itself:

• a = 5   ⇒   5b = 20   ⇒   b = 4

There are 2 multiples of 4 to check. The first one gives the same result as a = 5.

• b = 8   ⇒   4a = 40   ⇒   a = 10

but a must be 1-digit.

So among the 2-digit numbers, only 1 such number exists, n = 54.

Let n = abc be a 3-digit number satisfying the criterion. Then

n = 100a + 10b + c = 6(a + b + c)

100a + 10b + c = 6a + 6b + 6c

94a + 4b = 5c

c must be a number from 1-9, which means at most 5c = 5•9 = 45.

If we take the smallest possible values of a and b, we have

94•1 + 4•0 = 94

but the smallest value of 94a + 4b is larger than the largest value of 5c.

So there are no such 3-digit numbers.

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6 ft^2

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2 years ago
What is -14 divided by -2
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3 years ago
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Answer:

The answer is $10.80

Step-by-step explanation:

It goes up $0.70 every mile and the flat fee is $3.10.

The equation 3.1 + 0.7(x) when x is the number of miles traveled.

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3 years ago
the half life of c14 is 5730 years. Suppose that wood found at an archeological excavation site contains about 35% as much C14 a
Furkat [3]

Answer:

The wood was cut approximately 8679 years ago.

Step-by-step explanation:

At first we assume that examination occured in 2020. The decay of radioactive isotopes are represented by the following ordinary differential equation:

\frac{dm}{dt} = -\frac{m}{\tau} (Eq. 1)

Where:

\frac{dm}{dt} - First derivative of mass in time, measured in miligrams per year.

\tau - Time constant, measured in years.

m - Mass of the radioactive isotope, measured in miligrams.

Now we obtain the solution of this differential equation:

\int {\frac{dm}{m} } = -\frac{1}{\tau}\int dt

\ln m = -\frac{1}{\tau} + C

m(t) = m_{o}\cdot e^{-\frac{t}{\tau} } (Eq. 2)

Where:

m_{o} - Initial mass of isotope, measured in miligrams.

t - Time, measured in years.

And time is cleared within the equation:

t = -\tau \cdot \ln \left[\frac{m(t)}{m_{o}} \right]

Then, time constant can be found as a function of half-life:

\tau = \frac{t_{1/2}}{\ln 2} (Eq. 3)

If we know that t_{1/2} = 5730\,yr and \frac{m(t)}{m_{o}} = 0.35, then:

\tau = \frac{5730\,yr}{\ln 2}

\tau \approx 8266.643\,yr

t = -(8266.643\,yr)\cdot \ln 0.35

t \approx 8678.505\,yr

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3 years ago
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She needs to do a certain remaining hours of credit which can be represented by c and only c without any other coefficient.

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