Solve the problem.
-Rose's work ID is based on her last name, Smith, and her code number is $19 - 13 - 9 - 20 - 8$, written as 19139208. If Jen's ID is an 8-digit palindrome, say a - b - cd - e - $\mathrm { f } - \mathrm { gh }$, which satisfies the given conditions, what is Jen's last name?
$\begin{array} { l }
2 ( a + h ) = d + e ; \
a \times b \times h = d ; \
b + c = a ; \
a \neq b \neq d ;
\end{array}$
the sum of the digits is 16 ;
$c = b ; e = 4 \text {; }$
Jen's last name starts with a letter in the first half of the alphabet.
A) Andal
B) Rundal
C) Mandal
D) Bandal

Question

Solve the problem.
-Rose's work ID is based on her last name, Smith, and her code number is $19 - 13 - 9 - 20 - 8$, written as 19139208. If Jen's ID is an 8-digit palindrome, say a - b - cd - e - $\mathrm { f } - \mathrm { gh }$, which satisfies the given conditions, what is Jen's last name?
$\begin{array} { l } 
2 ( a + h ) = d + e ; \
a \times b \times h = d ; \
b + c = a ; \
a \neq b \neq d ;
\end{array}$
the sum of the digits is 16 ;
$c = b ; e = 4 \text {; }$
Jen's last name starts with a letter in the first half of the alphabet.
A) Andal
B) Rundal
C) Mandal
D) Bandal

Quizplus · Accepted Answer

Given the conditions and the sum of the digits being 16, we can deduce the following:- $c = b$ and $e = 4$, so $d + e = d + 4$.- $2(a + h) = d + 4$, and since $a \times b \times h = d$, we can infer that $a, b, h$, and $d$ must be small numbers to keep the sum of all digits to 16.- $b + c = a$, and since $c = b$, it means $2b = a$.Given $e = 4$ and the sum of all digits is 16, we can start by assuming smaller values for $a, b, h$, and $d$ that satisfy the equations. Let's assume $b = 1$ (since $a$ must be double $b$ and we aim to keep numbers small), then $c = 1$, and $a = 2$. This gives us $2 + 1 + 1 + 4 = 8$, leaving 8 for $d, h$, and the second occurrence of $a$. Given $a \times b \times h = d$, and $a = 2$, $b = 1$, if we choose $h = 2$ to maintain the palindrome structure, $d$ would then be $2 \times 1 \times 2 = 4$, which matches our requirement for $d$. So, the ID could be $2 - 1 - 1 - 4 - 4 - 1 - 1 - 2$, which sums to 16 and satisfies all given conditions. The corresponding letters to the numbers $2$ and $1$ are B and A, respectively, making the last name start with "Ba". Among the options, only "Bandal" starts with "Ba" and is in the first half of the alphabet, making it the correct choice.

Solve the Problem 19−13−9−20−819 - 13 - 9 - 20 - 819−13−9−20−8

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