# Mathcount Mixed Practice Level 2 Exams

These are some sample problems with their solutions from this course.

Example The floor of a square room is covered with congruent square tiles. The diagonals of the room are drawn across the floor, and the two diagonals intersect a total of 9 tiles. How many tiles are on the floor?
Solution Since we have square tiles filling up a square, we're talking about n*n or n^2 tiles. The first diagonal will go through the first tile in the first row, the second tile in the second row and so on. The second diagonal will go through the nth diagonal in the first row all the way to the first tile in the nth row. If n is odd, the diagonals will go through the same tile and that tile will be the middle tile in the middle row. If n is even, the diagonals will not go through the same tile nand the number of tiles touched by the diagonal will be even. But we're told the total number of tiles is 9, which is odd. 2n - 1 = 9, 2n = 10, n = 5 so the total number of tiles on the floor are n^2 = 5*5 = 25

Problem Noah wants to fill in the two blanks in the numeral 5__ 1__ 2 to create a five-digit positive integer that is divisible by 6. What is the greatest five-digit multiple of 6 that he can create?
Answer An integer is divisible by 6 if and only if it is divisible by 2 and by 3. An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6 or 8, and the number in question has units digit 2, so it is divisible by 2. An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sum of the three known digits is 5 + 1 + 2 = 8, which is 1 short of being a multiple of 3. Therefore, the two undetermined digits must add to a multiple of 3 plus 1; because digits cannot exceed 9, the sum of the two undetermined digits cannot exceed 18. As a result, the two digits in question can have of sum of 1, 4, 7, 10, 13 or 16. The desire is for the maximum value, which is 16 for the sum. A 9 in the left blank and a 7 in the right blank yields the greatest result, 59,172.

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Level 2 2001

• MP #2.1.1
• MP #2.1.2
• MP #2.1.3
• MP #2.1.4

Level 2 2002

• MP #2.2.1
• MP #2.2.2
• MP #2.2.3
• MP #2.2.4

Level 2 2003

• MP #2.3.1
• MP #2.3.2
• MP #2.3.3
• MP #2.3.4

Level 2 2004

• MP #2.4.1
• MP #2.4.2
• MP #2.4.3
• MP #2.4.4

Level 2 2005

• MP #2.5.1
• MP #2.5.2
• MP #2.5.3
• MP #2.5.4

Level 2 2006

• MP #2.6.1
• MP #2.6.2
• MP #2.6.3
• MP #2.6.4

Level 2 2007

• MP #2.7.1
• MP #2.7.2
• MP #2.7.3
• MP #2.7.4

Level 2 2008

• MP #2.8.1
• MP #2.8.2
• MP #2.8.3
• MP #2.8.4

Level 2 2009

• MP #2.9.1
• MP #2.9.2
• MP #2.9.3
• MP #2.9.4

Level 2 2010

• MP #2.10.1
• MP #2.10.2
• MP #2.10.3
• MP #2.10.4

Level 2 2011

• MP #2.11.1
• MP #2.11.2
• MP #2.11.3
• MP #2.11.4

Level 2 2012

• MP #2.12.1
• MP #2.12.2
• MP #2.12.3
• MP #2.12.4

Level 2 2013

• MP #2.13.1
• MP #2.13.2
• MP #2.13.3
• MP #2.13.4

Level 2 2014

• MP #2.14.1
• MP #2.14.2
• MP #2.14.3
• MP #2.14.4

Level 2 2015

• MP #2.15.1
• MP #2.15.2
• MP #2.15.3
• MP #2.15.4

Level 2 2016

• MP #2.16.1
• MP #2.16.2
• MP #2.16.3
• MP #2.16.4

Level 2 2017

• MP #2.17.1
• MP #2.17.2
• MP #2.17.3
• MP #2.17.4

Level 2 2018

• MP #2.18.1
• MP #2.18.2
• MP #2.18.3
• MP #2.18.4

Level 2 2019

• MP #2.19.1
• MP #2.19.2
• MP #2.19.3
• MP #2.19.4

Level 2 2020

• MP #2.20.1
• MP #2.20.2
• MP #2.20.3
• MP #2.20.4