Digits to show that 241 is not a perfect square
Any way to show that 241 is not a perfect square using digital rules??
We know that:
All squares end in 0, 1, 4, 5, 6, or 9. This rules out both 2 and 3. The smallest number that cannot be ruled out is 5.
Final 2 digits
Using E for even and Q for odd (an O cannot be told from a 0,) squares must end in 00, E1, E4, 25, Q6, or E9. This rules out 5, which ends in 05. The smallest number that cannot be ruled out as a perfect square using this rule is 21.
Final 3 digits
How about 3-digit endings?? Squares that end in 00 must end in 000, 100, 400, 500, 600, or 900. For 5, there's 025, 225, and 625. No 3-digit endings ending in 4 or 6 besides those that have already been ruled out by 2-digit endings can be ruled out. For 1 and 9, the rule is that the last 2 digits before the final digit must be a multiple of 4. This rules out 21, becuase the 3-digit endings allowed are 121, 321, 521, 721, and 921. The smallest number that cannot be ruled out using the rules above is now 24.
Final 4+ digits
A theorem used for finding groups of 4 or more digits squares cannot end in is:
If n can be ruled out as a perfect square by its final d digits, then 4n and 25n can be ruled out by their final n+2 digits.
This can rule out 24. The 4-digit endings for a square that ends in 024 are 1024, 3024, 5024, 7024, and 9024. The smallest number that cannot be ruled out is 41.
Now, there don't appear to be any additional rules for groups of final digits that can be used to rule out squares ending in 1 or 9. Thus showing 41 is not a perfect square is like showing 10 is not a multiple of 3 in that you can't use groups of final digits. You can, however, use the digital root. The digital root of a square must be 1, 4, 7, or 9, and the digital root of 41 is 5. All numbers that are not perfect squares from 2 to 240 can be ruled out using one of these methods. However, 241 still cannot be ruled out. Any additional digit-based rule anyone has?? Georgia guy 20:46, 24 February 2006 (UTC)
Is this going to rule out 241 as a perfect square??
It says at Divisor that you can decide a number's divisiblity by 7 by looking at the last digit and adding it to 3 times the number made out of all the other digits. Thus, 72 + 1 = 73, then 21 + 3 = 24, then 6 + 4 = 10, then 3 + 0 = 3. This results in 3 for 241. Any number the final digit has to be for a perfect square?? Georgia guy 22:31, 24 February 2006 (UTC)
The answer appears to be yes. The final digit for 241 is 3, but the final digit for a perfect square has to be 1, 2, 4, 7, 8, or 9. It finally worked! Georgia guy 22:33, 24 February 2006 (UTC)