# Wrong, yet instructive for multiplication and division The math teacher gave the student the multiplication of two numbers, of which the multiplier is greater than the multiplier by 202 unity. After the multiplication was done, the teacher asked to check them by dividing the found product by the multiplier. The obtained quotient is 288 and the rest is left 67; it follows, that the multiplication was done incorrectly.

Having found the error, the student admits it:
— W dodawaniu poszczególnych iloczynów cząstkowych obliczyłem o jedną jedynkę mniej.
— Tu nie chodzi o jedynkę, but by a thousand, which was abandoned by you - the teacher corrected.

On the basis of the above, let's find both numbers to be multiplied.

Product of multiplication 288 by the multiplier is o 1000 + 67 = 1067 less than the exact result of the multiplication.

In other words, multiplier multiplied by 288 and enlarged by 1067 is equal to the product sought. Hence it follows, that 1067 is completely divided by the multiplier. This multiplier must be greater than the rest, what fell out of the division when checking, it is greater than 67. We break down the number 1067 on factors: 1067 = 11 • 97. Hence the final conclusion, that the multiplier must be equal 97. Then the multiplier is 97 + 202 = 299, the exact product is 29 003, and the erroneous product found by the student was 28 003.

Here's another, similar example: The teacher gave the student a division of two numbers. The student received the quotient 57 and finally 52. He made the attempt by multiplying the quotient by the divisor and adding the remainder. He then received a number 17 380, but this number was not equal to the brave. The student's mistake was that, that when multiplication, the student read the second digit on the right in the divisor as 0, and it was indeed 6. What numbers did the teacher give the student?

The answer is: 20 800 divide by 364. But how to come to this answer?