SOLUTION: Hello. I am stuck with this question and have no idea how to start it. Here it is: A palindromic number is one which reads the same backwards as forwards. (a) Find a three-digit

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Question 1164864: Hello. I am stuck with this question and have no idea how to start it. Here it is:
A palindromic number is one which reads the same backwards as forwards.
(a) Find a three-digit palindromic number which is exactly 19 times the
sum of its digits.
(b) Find all four-digit palindromic numbers which are exactly 352 times
the sum of their digits.
Thanks!
Found 2 solutions by Alan3354, MathTherapy:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
A palindromic number is one which reads the same backwards as forwards.
(a) Find a three-digit palindromic number which is exactly 19 times the
sum of its digits.
-------------
It's a multiple of 19, and > 100.
Just find it by trying all multiples of 19.
---
114, 133, 152, etc.
===================
(b) Find all four-digit palindromic numbers which are exactly 352 times
the sum of their digits.
It's a multiple of 352, and > 1000.
1056, 1408, 1760, etc
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
Hello. I am stuck with this question and have no idea how to start it. Here it is:
A palindromic number is one which reads the same backwards as forwards.
(a) Find a three-digit palindromic number which is exactly 19 times the
sum of its digits.
(b) Find all four-digit palindromic numbers which are exactly 352 times
the sum of their digits.
Thanks!
(a)  A 3-digit palindrome will have the same hundreds and units digits

     Let hundreds/units and tens digits be H, and M, respectively

     Then the number is: 100H + 10M + H

     Since the number is 19 times the sum of its digits, we get: 100H + 10M + H = 19(H + M + H)

     101H + 10M = 19(2H + M)

     101H + 10M = 38H + 19M

     101H - 38H = 19M - 10M

     63H = 9M =====> 

     M is a DIGIT, so H, or hundreds/units digit can ONLY be 1. Therefore, M = 7(1) = 7

    



(b)  A 4-digit palindrome will have the same thousands and units digits, and the same hundreds and tens digits

     Let thousands/units and hundreds/tens digits be T, and H, respectively

     Then the number is: 1,000T + 100H + 10H + T, or 1,001T + 110H

     Since the number is 352 times the sum of its digits, we get: 1,001T + 110H = 352(T + H + H + T) 

     1,001T + 110H = 352(2T + 2H)

     1,001T + 110H = 704T + 704H

     1,001T - 704T = 704H - 110H

     297T = 594H =====> 

     H is a DIGIT, and can ONLY be 1, 2, 3, or 4. Therefore:   

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