1 Geometry Question

[MathStudent1999]

7. A block of wood in the shape of a right rectangular prism has a height of 6cm, a width of M, and a length of N, where M & N are integers and M>N. The outside of the block is painted and the block is cut into 6MN cubes. Exactly half of the cubes have no paint on them. What is the least possible value for N?

I did: 6MN-[4(M-2)(N-2)]=[4(M-2)(N-2)]. 6MN = 8(M-2)(N-2). 3MN = 4(M-2)(N-2). 3MN = 4(MN -4M - 4N +4). 3MN = 4MN - 4M - 4N +16. MN-4M-4N+16 = 0. (M-4)(N-4)=0

Guessing and Checking, the smallest value for N, that matched all the conditions would be 5, but there should be an easier way to do this. Can someone check if this answer is right and/or see if there is a easier way to do this?

 

[MathStudent1999]

I did: 6MN-[4(M-2)(N-2)]=[4(M-2)(N-2)]. 6MN = 8(M-2)(N-2). 3MN = 4(M-2)(N-2). 3MN = 4(MN -4M - 4N +4). 3MN = 4MN - 4M - 4N +16. MN-4M-4N+16 = 0. (M-4)(N-4)=0

There is a error in this that I just found. 3MN = 4(MN -2M - 2N +4) not 4(MN -4M - 4N +4). This means that MN-8N-8M+16 = 0 This equation goes nowhere. Can you tell me what to do next?

 

[soroban]

Hello, MathStudent1999!

7. A block of wood in the shape of a right rectangular prism
has a height of 6 cm, a width of M, and a length of N,
where M and N are integers and M > N.

The outside of the block is painted and the block is cut into 6MN cubes.
Exactly half of the cubes have no paint on them.
What is the least possible value for N?

The interior of the prism has: .4(M - 2)(N - 2) cubes.

We have: . 4(M - 2)(N - 2) .= .(1/2)(6MN)

. . . . 4(MN - 2M - 2N + 4) .= .3MN

. . . . .4MN - 8M - 8N + 16 .= .3MN

. . . . . . . . . . . . . MN - 8N .= .8M - 16

. . . . . . . . . . . . . (M - 8)N .= .8(M - 2)

. . . . . . . . . . . . . . . . . . . . . . .8(M - 2)
. . . . . . . . . . . . . . . . . . N .= .----------
. . . . . . . . . . . . . . . . . . . . . . . .M - 8


The least M is: .M = 9 . → . N = 56.

But they asked for the least N . . . and note that: M > N.


We find that the problem is satisfied when: .M = 56, N = 9.

 

[MathStudent1999]

Hello, MathStudent1999!


The interior of the prism has: .4(M - 2)(N - 2) cubes.

We have: . 4(M - 2)(N - 2) .= .(1/2)(6MN)

. . . . 4(MN - 2M - 2N + 4) .= .3MN

. . . . .4MN - 8M - 8N + 16 .= .3MN

. . . . . . . . . . . . . MN - 8N .= .8M - 16

. . . . . . . . . . . . . (M - 8)N .= .8(M - 2)

. . . . . . . . . . . . . . . . . . . . . . .8(M - 2)
. . . . . . . . . . . . . . . . . . N .= .----------
. . . . . . . . . . . . . . . . . . . . . . . .M - 8


The least M is: .M = 9 . → . N = 56.

But they asked for the least N . . . and note that: M > N.


We find that the problem is satisfied when: .M = 56, N = 9.

Right, thanks!