+743 How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$
+1926 The first number above 1000 that is divisble by 47 is 1034. Thus, the smallest number that fits the list is 1037.
The largest number under 2000 divisble by 47 is 1974, so 1977 is the last number in our list.
Now, notice that we just have an arithmetic sequences. We have
\(1037, 1084, 1131, ... 1977\)
Subtracting every number in the list by 3, we get
\(1034, 1081, 1128, ...1974\)
Divididing every number in the list by 47, we get
\(22, 23, 24,...42\)
Subtracting 21 from every term in the sequences, we have
\(1,2,3,4...21\)
There are 21 numbers in this list.
Thus, there are 21 numbers that leave a remainder of 3 when divided by 47.
So our answer is 21.
Thanks! :)
~NTS
