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!
<pre><b>(a)</b>  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 =====> {{{matrix(2,3, 63H/9, "=", M, 7H, "=", M)}}}
     M is a DIGIT, so H, or hundreds/units digit can ONLY be 1. Therefore, M = 7(1) = 7
    {{{highlight_green(matrix(1,2, "Palindrome:", 171))}}}

<b>(b)</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 =====> {{{matrix(2,3, T, "=", 594H/297, T, "=", 2H)}}}
     H is a DIGIT, and can ONLY be 1, 2, 3, or 4. Therefore:  {{{ highlight_ green(matrix(4,14, If, H, "=", 1, then, T, "=", 2, and, the, 4-digit, palindrome, "is:", "2,112",
If, H, "=", 2, then, T, "=", 4, and, the, 4-digit, palindrome, "is:", "4,224", If, H, "=", 3, then, T, "=", 6, and, the, 4-digit, palindrome, "is:", "6,336", If, H, "=", 4, then, T, "=", 8, and, the, 4-digit, palindrome, "is:", "8,448"))}}}