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CEMC Gauss (Grade 8), 2018

Algebra Difficulty 1.6 Junior Find the answer Canada

The volume of a rectangular prism is 30 cm $^3$ . The length of the prism is doubled, the width is tripled, and the height is divided by four. The volume of the new prism is

Pick one

Solution

For the product of two integers to equal 1, the two integers must both equal 1 or must both equal $-1$ .

Similarly, if the product of six integers is equal to 1, then each of the six integers must equal 1 or $-1$ .

For the product of six integers, each of which is equal to 1 or $-1$ , to equal 1, the number of $-1$ s must be even, because an odd number of $-1$ s would give a product that is negative.

That is, there must be zero, two, four or six $-1$ s among the six integers.

We summarize these four possibilities in the table below.

Number of $-1$ s	Product of the six integers	Sum of the six integers
0	$(1)(1)(1)(1)(1)(1)=1$	$1+1+1+1+1+1=6$
2	$(-1)(-1)(1)(1)(1)(1)=1$	$(-1)+(-1)+1+1+1+1=2$
4	$(-1)(-1)(-1)(-1)(1)(1)=1$	$(-1)+(-1)+(-1)+(-1)+1+1=-2$
6	$(-1)(-1)(-1)(-1)(-1)(-1)=1$	$(-1)+(-1)+(-1)+(-1)+(-1)+(-1)=-6$

Of the answers given, the sum of such a group of six integers cannot equal 0.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.