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.
+4622 1): We can use the Chinese Remainder Theorem.
I can post a solution if you want.
+129899 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
10003 to 22223 = [27 to 80] base10
The integers in base 6 can range from
106 to 556 = [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
