Try this beautiful Problem based on Divisibility Rule from Math Kangaroo (Benjamin) 2020.
We say that a three-digit number is balanced if the middle digit is the arithmetic mean of the other two digits. How many balanced numbers are divisible by 18 ?
Arithmetic
Divisibility Rule
Arithmetic Mean
Algebra by Gelfand
Math Kangaroo (Benjamin), 2020
6
Let the number be $abc.$
By the given condition of arithmetic mean we get,
$$\frac{a+c}{2}=b$$.
Now from here can I conclude $a, c$ are of the same parity?
Given that the number is divisible by $18$
$\Rightarrow$ Given number must be divisible by both $2$ and $9$.
Apply divisibility rule of $9$ to guess the values of $a, b, c$.
Here the values of $3b$ are $9, 18, 27$.
$\Rightarrow$ the values of $2b$ are $6, 12, 18$.
Now guess the values of $a+c$?
And then find the possible pair of $(a, c)$?
The possible values of $(a, c)$ are $(2,4)$, $(4,2)$, $(6,0)$, $(4,8)$, $(6,6)$, $(8,4)$.
So, how many possible numbers are there?
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