1) Determine the smallest non-negative integer that satisfies the congruences:

a = 2 (mod 3)

a = 4 (mod 5)

a = 6 (mod 7)

a = 8 (mod 9)

2) What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.

Q3 deleted.

Just ask one question per post. Thanks.

Guest May 26, 2019

#1**+1 **

1): We can use the Chinese Remainder Theorem.

I can post a solution if you want.

tertre May 26, 2019

#2**+1 **

2) What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.

The integers in base 3 can range from

1000_{3} to 2222_{3 }= [27 to 80] base10

The integers in base 6 can range from

10_{6} to 55_{6} = [6 to 35] base 10

So the common integers are 27, 28, 29 , 30 , 31, 32, 33 , 34 and 35

Their average = [ 4(62) + 31] / 9 = 31

CPhill May 26, 2019