Question

Please help!!

Answer 1

Brandon spends his afternoon picking apples from an orchard. He notices that if he groups the apples he picked by 5's or 6's, then he ends up with 3 left over. If he groups by 7's or 8's, he has 5 left over. If Brandon knows that he picked between 1000 and 2000 apples, how many did he pick?

Answer 2

It all boils down to

\[N\equiv 3\ (mod 5)\]
\[N\equiv 3\ (mod 6)\]
\[N\equiv 5\ (mod 7)\]
\[N\equiv 3\ (mod 8)\]

Solve. I think. But I don't know how to do those! Could someone give the answer and then explain?

Answer 3

This question looks very similar to the ones we were doing last time...

Answer 4

Yep.

Answer 5

I have a lot on the topic.

Answer 6

I guess your last equation should be \(N≡5 \mod 8 \)?

Answer 7

oops. yeah.

Answer 8

I don't see a clever way of doing it. As far as I can see, you'll just have to solve the first pair, deal with the next pair and finally solve the last two equations. Do you still remember the chinese remainder theorem from last time?

Answer 9

no. I never really understood how to do it.

Answer 10

I do remeber talking about it though

Answer 11

Now this question is practically identical to the other one :)

http://openstudy.com/study#/updates/52195471e4b06211a67cafc0

You even have the same number of equations!

Answer 12

ok. I need to leave now. its and...emergency. real life sorry!!!

Answer 13

crt again
good luck

Answer 14

got to be odd, right? and a multiple of 3.

6x+3 = n

so (2x+1)*3 = n

n=3mod5 => 2x+1 =1 mod 5

n=5mod7 => 2x+1 = 4 mod 7

n=5 mod 8 => 2x+1 = 7 mod 8

Answer 15

I don't know...just solve the equations?

Answer 16

31 fits the bill...

31= 1mod 5
31 = 4 mod 7
31 = 7 mod 8

so n should be 3*(c*31), yeah?

Answer 17

wait so 31 works. it works?

Answer 18

wait no. It's 31 times something.

Answer 19

yeah but something is off...

Answer 20

there are four equations?

Answer 21

the number is 1293... but how?

Answer 22

How'd you get that?

Answer 23

It;s right. Wait I've got the solution.

Answer 24

2x+1= 5 mod 6, but why?


oh... excel

Answer 25

2x+2 = 6 mod 6 so 2x+1=5 mod 6.

because 3(2x+1) + 3 = 0 mod 6

Answer 26

that makes 2x+1 = 431

Answer 27

3*431 = 1293

Answer 28

Let n be the number of apples that Brandon picked. Then \[n \equiv 3 \pmod{5} \]\[n \equiv 3 \pmod{6} \]\[n \equiv 5 \pmod{7} \]\[n \equiv 5 \pmod{8}\]...

From the first two congruences, \[n \equiv 3 \pmod{30}. \] From the last two congruences, \[n \equiv 5 \pmod{56}. \] Hence, \[n = 30a + 3 = 56b + 5 \] for some integers a and b, which simplifies as \[30a = 56b + 2, or 15a = 28b + 1 \].

One solution is a = 15 and b = 8. Subtracting 225 from both sides of the equation \[15a = 28b + 1 \], we get \[15a - 225 = 28b - 224 \], which factors as \[15(a - 15) = 28(b - 8) \].

This equation tells us that 15(a - 15) is divisible by 28, but 15 is relatively prime to 28, so a - 15 is divisible by 28. Then a - 15 = 28k for some integer k, so a = 28k + 15, and n = 30a + 3 = 30(28k + 15) + 3 = 840k + 453.

The only number of this form between 1000 and 2000 is 1293.

Answer 29

what class are you taking? you have the best questions! i want to take your class.

Answer 30

Just Number Theory.

Answer 31

As in the class is called Number Theory. I sometimes ask questions for my Counting and Probability class too.

Answer 32

http://openstudy.com/study#/updates/521d3a27e4b06211a67dc352

Answer 33

nm. I got te answer

Answer 34

what school do you go to?

Answer 35

Klondike Middle School

Answer 36

wow!

Answer 37

heh. It's an online class. :)