Problem of the Day


Let a,b,c, and d represent real numbers. Is c^{abcd}=1?

(1) |a|>|b|>|c|>|d|

(2) c^d>3

 

Reveal Answer

Answer

C. Both statements together are sufficient, but neither statement alone is sufficient.

See the Solution

Solution

[latexpage]

We are trying to determine if $c^{abcd}=1$. This could be true if $c=1$ or if $abcd=0$.

Statement 1: $|a|>|b|>|c|>|d|$. This implies that $a,b,$ and $c$ are not equal to zero. Otherwise they could not possibly have a greater absolute value than some other number. However, $d$ could still be equal to zero. So, the statement could be true if $d=0$ or it could be false if, for example, $a=10, b=9, c=8, d=7$. INSUFFICIENT.

Statement 2: $c^d>3$. This implies that $c\neq1$ and $d\neq0$. The statement could be true if $a=0,b=5,c=4,d=3$ or false if $a=10,b=9,c=8,d=7$. INSUFFICIENT.

Statements 1&2: With both statements we can conclude that $c\neq1$ and $abcd\neq0$. Therefore $c^{abcd}\neq1$. SUFFICIENT.


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