PalindromesDate: 02/24/98 at 13:46:35 From: Anonymous Subject: Palindromes Two 4-digit palindromes are added to give a 5-digit palindrome N. What are the possible numbers for N? Date: 02/24/98 at 17:34:57 From: Doctor Sam Subject: Re: Palindromes Keshni, Suppose we represent the digits of the palindrome with letters. Your problem becomes: a b b a + c d d c ------------- e f g f e The largest possible four-digit number is 9999, which is less than 10,000. So abba + cddc must be less than 20,000. So e = 1. Now we know that e = 1, so the units digit of the sum is one. How can a+c give 1? Neither can be zero (since we cannot start a number with 0) so it must be that a+c = 11. Now if a+c = 11 in the units place, then a+c also gives 11 in the thousands place. What does that tell us about f? Either there is a carry from the b+d addition in the hundreds place or not. This gives two possibilities: NO CARRY from b+d CARRY from b+d a b b a a b b a + c d d c + c d d c ------------- ------------- 1 1 g f 1 1 2 g f 1 This gives us a value for f: (1) a b b a (2) a b b a + c d d c + c d d c ------------ ------------- 1 1 g 1 1 1 2 g 2 1 Now what? In case (1) there is no carry from b+d and a+c = 1. In this case the addition in the tens place is b+d+1 = 1 or 11. The 11 answer is not a possibility here because that would make b+d = 10 and we are assuming that there is no carry from b+d. So the only possibility is b+d = 0, which only happens if b = 0 and d = 0. In this case g = 0 as well and there are four possibilities: 2 0 0 2 3 0 0 3 4 0 0 4 5 0 0 5 + 9 0 0 9 + 8 0 0 8 + 7 0 0 7 + 6 0 0 6 --------- --------- --------- --------- 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 So in this case N = 11011 That leaves case (2) where b+d produces a carry. CARRY from b+d a b b a + c d d c ------------- 1 2 g f 1 Since this is a palindrome, f = 2. Since a+c = 11, we can now look at the addition in the tens place: b + d + 1 = 2 or 12 (since f = 2). case (2a): b+d+1=2 so b+d = 1. This can only happen if b = 0 and d = 1 or vice versa. This gives two more options: a 0 0 a a 1 1 a + c 1 1 c + c 0 0 c ------------- ------------- 1 2 1 2 1 1 2 1 2 1 So N = 12121 Finally we have case (2b): b+d+1 = 12 or b+d = 11. In this case g = 2 and so N = 12221. Hope that helps. -Doctor Sam, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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