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alekssr [168]
3 years ago
9

Prove that every integer greater than 7 can be written by using 3's and 5's only. that is, for every n > 7 there exist non-ne

gative integers tn, sn such that n = 3tn + 5sn.
Mathematics
1 answer:
Taya2010 [7]3 years ago
4 0
An integer may be a multiple of 3.
An integer may be 1 greater than a multiple of 3.
An integer may be 2 greater than a multiple of 3.

It is redundant to say an integer is 3 greater than a multiple of 3 (that's just a multiple of 3, we've got it covered).  Same for 4, 5, 6, 7...

Let's consider a number which is a multiple of 3.  Clearly, we can write 3+3+3+3+... until we reach the number.  It can be written as only 3's.

Let's consider a number which is 2 greater than a multiple of 3.  If we subtract 5 from that number, it'll be a multiple of 3.  That means we can write the number as 5+3+3+3+3+...  Of course, the number must be at least 8.

Let's consider a number which is 1 greater than a multiple of 3.  If we subtract 5 from that number, it'll be 2 greater than a multiple of 3.  If we subtract another 5, it'll be a multiple of 3.  That means we can write the number as 5+5+3+3+3+3+...  Of course, the number must be at least 13.

That's it.  We considered all the numbers.  We forgot 9, 10, 11, and 12, but these are easy peasy.

Beautiful question.
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Answer:

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4 0
2 years ago
Use the graph below to answer the following question: What is the average rate of chage from x = -4 to x = 1?
solong [7]

Answer:

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Step-by-step explanation:

8 0
3 years ago
Please help<br><br> 1) _____<br><br> 2) _____<br><br> 3) _____ <br><br> 4) ____<br><br> 5) ____
Luba_88 [7]

Answer:

1 B

2 C

3 E

4 D

5 A

Step-by-step explanation:

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7 0
2 years ago
Read 2 more answers
20 boxes of apples have been delivered to a restaurant. The number of apples in each box is counted. Complete the table below an
kirill [66]

Answer:

Frequency: 20

A.F: 650

Mean: 30

Step-by-step explanation:

Frequency and A.F were just sums of the values mentioned for each

The mean:

Definition: the average of a data.

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Answer: (28 x 2 + 29 x 4+ 30 x 8 + 31 x 4+ 32 x 2) divided by 20 = 30

Hope this helps

8 0
3 years ago
Sin (2x -9)= Cos (x +30) to solve for x
lukranit [14]
A graphing calculator shows four (4) zeros of
  f(x) = sin(2x -9°) -cos(x +30°)
in the range 0° ≤ x ≤ 360°

Solutions are x ∈ {23°, 129°, 143°, 263°).

_____
You can make use of a couple of trig identities to rearrange this equation.
  cos(x) = sin(x+90°)
  sin(a) -sin(b) = 2*cos((a+b)/2)*sin((a-b)/2)
So
  sin(2x -9°) -sin(x +120°) = 2*cos((3x +111°)/2)*sin((x -129°)/2) = 0
The cosine factor will be zero for
  (3x +111°)/2 = n*180° +90°
  3x -69° = n*360°
  x = 23° +n*120° . . . . . . for any integer n
The sine factor will be zero for
  (x -129°)/2 = n*180°
  x = 129° +n*360° . . . . . for any integer n

Combined, these solutions give the ones listed above in the range 0..360°.

4 0
3 years ago
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