(Palindrome)_8
Observe that 9 = ( 9 ) 1 0 is a palindrome in base 10, and 9 = ( 1 1 ) 8 is a palindrome base 8. In decimal notation, what is the next smallest integer that is a palindrome in both base 10 and base 8?
Details and assumptions
The "next smallest integer" refers to the smallest integer larger than 9 which satisfies the conditions.
The answer is 121.
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2 solutions
I want to suggest another method: First to figure out the number of digits in the palindrome in base 8: If the number is 2 digit, i.e. "aa" in base 8 it'll be 9 \times a in base 10. But since "a" ranges from 1 to 7, no such number is possible. If the number is 3 digit, i.e. "aba" in base 8, it'll be 65 \times a+8 \times b in base 10. Now if the number lies below 200 in base 10, then the first and last digits must be 1 so the only possibility is for a=1 and b=7 which gives 65 \times 1 + 8 \times 7 = 121 as the number in base 10. Sorry for writing it in the comments section. I accidentally missed to write in solution section.
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I like your method, however one doubt: you say if the number lies below 200 in base 10, then the first and the last digit must be 1. Why can't they be 2 or 3? For example, ( 2 0 2 ) 8 = ( 1 3 0 ) 1 0 or ( 3 0 3 ) 8 = ( 1 9 5 ) 1 0
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By saying "the first and last digits must be 1" I meant for the quantity 65 \times a+8 \times b, which is already in base 10, so we don't convert it into base 10 again. In your examples, you are converting a below 200 base 10 number into base 8 but to fulfill the criteria the number has to be in base 10 itself [so it will always be below 200 of course]. I hope I made it clear....
Ok. I mixed up with the 'aba' thing. Thanks!
How did you figure out the base 8 of the numbers?
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It counts up by 8, once it reaches 8, the tens digit increases by 1
1 2 3 4 5 6 7
10 11 12 13 14 15 16 17
20 21 22....
so on the ones digit never surpasses 7
Additionally, what is the tens digit in base ten, is the "eights digit" in base 8
I did base 10 to base 8 conversion by figuring out that
n base 8 is: 10*(the number-(the reminder when you divide by 8)) divided by eight+ the reminder when you divide by 8.
the reminder can be expressed as n (mod 8).
so if n is base 10
edit: hmm I got some kind of error here, if you want to use this formula, use the simplified one at the bottom
n base 8 = 10*((n - (n (mod 8)))/8) - n (mod 8)
(copy it to wolfram alpha if you want to see it in a more decent form)
which can be later simplified to
(5*n- (n(mod 8)))/4
First thing to do is whip out your conversions calculator. Now we can just go through all of the palindromes. We see that 121 fits the conditions, in base 8 it is 171.
22 base 8 = 18 33 base 8 = 27 So on 36, 45, 54, 63 there are no more palindromes. Next three-digit palindrome, 101 base 8 = 65 111 base 8 = 73 So on, 81, 89, 97, 105, 113, 121 We found 121 and 171 are palindromes.