4. Calculate the arithmetic mean of all multiples of the number 12 that are of the form $\overline{3 a 8 b}$. Which of the obtained multiples should be excluded so that the arithmetic mean of the remaining multiples is 50 greater than the arithmetic mean of all multiples?
4. Calculate the arithmetic mean of all multiples of the number 12 that are of the form $\overline{3 a 8 b}$. Which of the obtained multiples should be omitted so that the arithmetic mean of the remaining multiples is 50 greater than the arithmetic mean of all multiples?

## Solution.

A number is divisible by 12 if it is divisible by both 4 and 3.

A number is divisible by 4 if its last two digits are divisible by 4, so the digit $b$ can be 0, 4, or 8. 1 POINT

For $b=0$, the number is of the form $\overline{3 a 80}$.

$3+a+8+0=a+11$ so the digit $a$ can be 1, 4, or 7, and

the resulting numbers are $3180, 3480, 3780$.

For $b=4$, the number is of the form $\overline{3 a 84}$.

$3+a+8+4=a+15$ so the digit $a$ can be $0, 3, 6$, or 9, and

the resulting numbers are $3084, 3384, 3684, 3984$.

1 POINT

For $b=8$, the number is of the form $\overline{3 a 88}$.

$3+a+8+8=a+19$ so the digit $a$ can be 2, 5, or 8, and

the resulting numbers are $3288, 3588, 3888$.

1 POINT

The arithmetic mean of the numbers is

$(3180+3480+3780+3084+3384+3684+3984+3288+3588+3888): 10=$ $35340: 10=3534$.

Let $x$ be the number that needs to be omitted. Then it holds that

$(35340-x): 9=3584$

2 POINTS

$35340-x=3584 \cdot 9$

$35340-x=32256 \quad 1$ POINT

$x=35340-32256 \quad 1$ POINT

$x=3084$

The number 3084 should be omitted.

Note: If it is clear from the solution that the desired solution is the number 3084, a point should not be deducted for the lack of a written answer.