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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 ?

- 2
- 3
- 6
- 9
- 18

Arithmetic

Divisibility Rule

Arithmetic Mean

Suggested Reading

Source of the Problem

Answer

Algebra by Gelfand

Math Kangaroo (Benjamin), 2020

6

Hint 1

Hint 2

Hint 3

Hint 4

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?

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 ?

- 2
- 3
- 6
- 9
- 18

Arithmetic

Divisibility Rule

Arithmetic Mean

Suggested Reading

Source of the Problem

Answer

Algebra by Gelfand

Math Kangaroo (Benjamin), 2020

6

Hint 1

Hint 2

Hint 3

Hint 4

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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