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## Question

A tailor with a real faney for cutting cloth has ten pieces of material. He decides to cut some of this material into 10 pieces each. He then cuts some of those resulting pieces into ten pieces each. He continues this way until he finally tires and stops. He counts the number of cloth pieces; after a few minutes he determines that there are 1984 pieces of cloth. Show that his count must have been incorrect .

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If he keeps cutting each cloth into ten pieces then the result must be a multiple of ten.

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Operation of cutting one piece into 10 pieces increases number of pieces by 9 thus preserving number of pieces modulo 9. A finite number of such operations will preserve it too. But 10 and 1984 are not equal modulo 9.

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So initially he has 10 pieces.

Once he cuts a piece he will have 9 (uncut in this step) + 10 new pieces = 19 = 10+1*9

Next time he has 18(uncut in this step) +10 new = 28 pieces = 10+2*9

So whenever he cuts a piece into 10, he has 10+n*9 pieces.

But 1984 is not equal to 10+n*9 for any n.

Q.E.D.

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A tailor with a real faney for cutting cloth has ten pieces of material. He decides to cut some of this material into 10 pieces each. He then cuts some of those resulting pieces into ten pieces each. He continues this way until he finally tires and stops. He counts the number of cloth pieces; after a few minutes he determines that there are 1984 pieces of cloth. Show that his count must have been incorrect .

The tailors pieces can be modeled by the equation;

N=(10-A)+10(A-B)+10(10(A-B)-C)+10(10(10(A-B)-C)-D)

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shouldn't it be a multiple of 10 because if he cuts it in 10 each time then he'd be wrong?

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