One Too Many Times

Number Theory Level 3

The product of 11 integers is equal to 11.

What is the minimum number of times the integer "1" appears in these 11 numbers?

0 1 2 3

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Anandmay Patel
Oct 14, 2016

Relevant wiki: Integers - Problem Solving

Probably one would think the answer should be 10 times.But notice that we are free to \text{Probably one would think the answer should be 10 times.But notice that we are free to} use numbers from the whole set of integers! \text{use numbers from the whole set of integers!}

That includes \text{That includes} − 1 ! -1!

So we can do like this: \text{So we can do like this:}

− 1 × − 1 × − 1 × − 1 × − 1 × − 1 × − 1 × − 1 × − 1 × − 1 ⏟ ten times × 11 = ( − 1 ) 10 × 11 = 1 × 11 = 11 , \underbrace{ -1 \times -1 \times -1 \times -1 \times -1 \times -1 \times -1 \times -1 \times-1 \times -1}_{\text{ten times}} \times 11 = (-1)^{10} \times 11 = 1\times11= 11 ,

where no 1 appears.

9 Helpful 0 Interesting 0 Brilliant 0 Confused

It should be (-1) * (-1) *(-1) * (-1) *(-1) * (-1) *(-1) * (-1) *(-1) * (-1) * 11 = 11

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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