SOLUTION: How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$

Algebra.Com
Question 1207599: How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$
Found 2 solutions by Edwin McCravy, mccravyedwin:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
This site only uses HTML, so don't use other notation.
Such an integer must be 3 more than a multiple of 47. They form an arithmetic
sequence, with common difference d=47

To find the first term, we first find the first multiple of 47 above 1000.

1000/47 = 21.27659574

So we know from that, that since the whole part is 21, that 21x47 is the last
multiple of 47 less than 1000, and that the first multiple of 47 above 1000 will
be 22x47=1034.

Add 3 to that and we get the first term, 







We require the terms to be less than or equal to 2000.



Solve that inequality and get





So the answer is the largest integer less than 21.4893617, which is 21.

Edwin

Answer by mccravyedwin(408)   (Show Source): You can put this solution on YOUR website!
It just occurred to me that we could have known the answer from that first
step:

1000/47 = 21.27659574

We could deduce the answer is 21, because there are just as many positive
multiples of 47 between 0 and 1000 as there are between 1000 and 2000, and
therefore just as many integers that are 3 more than a multiple of 47 as well.
That's because they are a long way from the endpoints of the intervals, 0, 1000,
and 2000.

Hey, don't turn in what I just wrote! I just wrote it to maybe help your
thinking about numbers.  LOL.

Edwin

RELATED QUESTIONS

How many different counting numbers will each leave a remainder of 5 when divided into... (answered by Boreal,greenestamps)
How many prime numbers between 8 and 60 leave a remainder of 2 when divided by... (answered by Alan3354,jim_thompson5910)
how many 3 digit numbers have a reminder of 7 when divided by 9, and a remainder of 2... (answered by ankor@dixie-net.com)
Please show me how to solve this problem " how many numbers are there between 1 and 150 (answered by mathmate)
list all the two-digit numbers which leave a remainder of 6 and when 171 is divided by... (answered by Fombitz)
how many two digit prime numbers have a remainder of 2 when divided by... (answered by solver91311)
How many two_digit whole numbers yield a remainder of 1 when divided by 10 and also yield (answered by LinnW)
There is a boy named Regan who collects baseball cards. He has between 2500 and 3000... (answered by Edwin McCravy)
For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when... (answered by ikleyn,math_tutor2020)