Six of the factors of 1221 are palindromes: {1,3,11,33,111,1221} but two are not: {37,407}
(A) What number under one million has the most palindrome factors?
Some numbers have only palindromes as factors. Examples include 88: {1,2,4,8,11,22,44,88} and any palindromic prime
(B) Find the smallest number with at least as many palindrome factors as part (A) but having only palindrome factors.
(In reply to
re(2): My thoughts on the difficulty of part (B) by Jer)
Here are a few prime palindromes containing only 1s and 0s (with no adjacent 1's):
101
101000010000101
10000010101000001
1000010101010100001
1001010100010101001
1010000101010000101
1010001001001000101
Consider 33: 4 factors {1,3,11,33}
Multiply by 101000010000101
Get 3333000330003333 which has 8 factors:
{1,
3,
11,
33,
101000010000101,
303000030000303,
1111000110001111,
3333000330003333}
(or just use 33 * 101)
So since 48884 has 5 digits, I think that means are large 10...01 number must have at least 4 zeros between each digit that is 1.
So 90 digits minimum? But since 18 1's is divisible by 3, we need one more 1,
so 95 digits minimum? (to get 2*18 = 36 factors, anyway)
Edited on March 16, 2022, 4:26 pm
|
|
Posted by Larry
on 2022-03-16 15:46:50 |
re(3): My thoughts on the difficulty of part (B)
