Is that Divisible?

Question text

A number 32a5b¯¯¯¯¯¯¯¯¯¯¯¯

is divisible by 11. Its last two digits form a number 5b¯¯¯¯¯ that can be exactly divided by 3. What is the largest 5

-digit number with this with these properties?

Your Answer:

Solution

Since 5b¯¯¯¯¯

is divisible by 3, b should be either 1, 4 or 7. If 32a5b¯¯¯¯¯¯¯¯¯¯¯¯ is to be divisible by 11, it must be the case that

3−2+a−5+b=a+b−4

is a multiple of 11. If b=1, then a=3; if b=4 then a=0; and if b=7, then a=8. The three 4-digit numbers that satisfy these properties are then 32351, 32054 and 32857. Thus, the largest 5-digit number with these properties is 32857

.

The correct answer is: 32857

 

My Solution (That above one was crappy)

 If 5b is divisible by 3, then b must be 1,4, or 7, because only for those values will 5+b be divisible by 3. If 32a5b is divisible by 11, we must have 3-2+a-5+b, or a+b-4, must be divisible by 11. Let's say b is 7. Then a has to be 8. So, the resulting number is 32857. But what if b is 4? Then a is 0, for the answer to be 32054. If b is 1, then a is 3, 32351. The largest is 32857.

Important:

Remember that abcdef is divisible by 11 ifandonlyif a-b+c-d+e is divisible by 11.

A Symmetrical Octagon

Problem 6 – Correct! – Score: 1 / 7 (27650)

Report an ErrorProblem:

Ellie wants to color two sides of a regular octagon green, two blue, two red, and two yellow such that each pair of opposite sides of the octagon have the same color. How many different patterns can she form? (Two patterns are considered identical if one can be rotated to form the other.)

Solution:

Suppose she first chooses a side to color green. Because two patterns are considered identical if one can be rotated to form the other, it doesn't matter which side she chooses. She can then spin her octagon so that this first colored side is on "top". Then, suppose she goes clockwise around the octagon to finish coloring. She can't use green because the other green side must be opposite the first side. So, she has  choices for the first side after the green side. She then has  colors remaining for the next side, and remaining for the side after that. At this point, she has four consecutive sides that are four different colors. The other four sides are then determined: each is the color of its opposite side. So, there are  possible colorings.

Hint(s):

Grab some crayons and try it!

Your Response(s):

• :( 2520

• :( 315

• :( 3

• :( 24

• :) 6
• Analysis: There is only 4 sides that matter, since the other four sides are mirrored. There are 4! ways to color the four sides, but we overcounted because of rotations. 4!/4 = 3! = 6.