Tau Numbers
The positive integer has divisors. If is a divisor of , then is a tau number. What is the product of tau numbers which have only odd digits?
The answer is 9.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The odd numbers don't have even divisors, so even tau numbers have only odd divisors. If a tau number have only odd digits, than it has got odd divisors. We know, that only square numbers have odd number of divisors, so we will find those square numbers which don't have even digits.
Let's look the last two digits. A square number's last digit should be 0, 1, 4, 5, 6, 9, so the possible endings are: 11, 15, 19, 31, 35, 39, 51, 55, 59, 71, 75, 79, 91, 95, 99. A number can have 0, 1, 2 or 3 remainder divided by four, so the square can have 0 or 1 remainder divided by four. But the 11, 15, 19, 31, 35, 39, 51, 55, 59, 71, 75, 79, 91, 95, 99 numbers have 3 remainder by four. The only chance to have a tau number with only odd digits, if the number don't have more than one digit. The one-digit square numbers are 1 and 9. These are tau numbers because 1 is divisor of 1 and 3 is divisor of 9.
The answer is 1 ∗ 9 = 9 .