Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem20

Video Transcription

Question number 20.  Roo wants to find a six-digit number, the sum of whose digits is even, and the product of whose digits is odd. Which of the following statements about such a number is correct?  Statement A says that either two or four of the digits are even.  However, we cannot have any of the digits being even because with just one even digit, the product of the digits would also be even. So we make the note that all digits are odd,  and we can go ahead and eliminate A from our choices.  B says that such a number cannot exist.  So let's think of an example.  Maybe that's the same number that Roo came up with.  111,111.  So we eliminate B. Here, the sum of the digits is even, and the product is odd. We also note that we used all of the same number, and there are  six, or an even number of odd numbers here,  which is enough to eliminate choice C.  Now, for D, which says that all six digits can be different, we go ahead and see if we can come up with such a number with all different odd numbers.  And it seems like we should have no trouble writing down 1, 3, 5, 7, 9, and 11, but now this is a seven-digit number.  And so D is also not possible, and by the process of elimination, we are left with E as the answer to number 20.

Video Summary

Roo seeks a six-digit number where the sum of its digits is even and the product is odd. Initially, Statement A suggests having two or four even digits, but this contradicts the requirement for an odd product. Statement B claims such a number can't exist, but example 111,111 fits the criteria, so B is incorrect. Statement C is ruled out since having six odd digits is possible. For Statement D, trying to use all different odd digits results in a seven-digit number, so D is also incorrect. By elimination, the correct conclusion is Statement E.

Keywords

six-digit number

even sum

odd product

contradiction

elimination