The Game of Cats and Mice

Number Theory Level 3

A number of cats got together and decided to kill between them 999919 mice. Every cat killed an equal number of mice.

How many cats do you think there were?

Oh by the way let me clarify just two points – it is not one cat killed the lot, because I have said 'Cats' and it is not 999919 cats each killed one mouse, because I have used the word 'mice'.

I can give you just one clue – Each cat killed more mice than there were cats


The answer is 991.

Sachin Sharma by Sachin Sharma , 32, India

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2 solutions

. .
Mar 2, 2021

Use the factor formula. ( a + b ) ( a − b ) = a 2 − b 2 → a 2 − b 2 = ( a + b ) ( a − b ) ⇒ 999919 = 100000 − 81 , and 81 = ± 9 , but, the number of cats, and mice cannot be negative, so + 9. Then, ( 1000 + 9 ) ( 1000 − 9 ) = 1009 × 991 = 999919 ⇒ 999919 = ( 1000 + 9 ) ( 1000 − 9 ) = 1009 × 991. And, the problem said that the number of mice > the number of cats, so the answer is 1009 > 991 → 991 \text { Use the factor formula. } ( a + b ) ( a - b ) = a ^ { 2 } - b ^ { 2 } \rightarrow a ^ { 2 } - b ^ { 2 } = ( a + b ) ( a - b ) \Rightarrow 999919 = 100000 - 81, \text { and } \sqrt { 81 } = \pm 9, \text { but, the number of cats, and mice cannot be negative, so } +9. \text { Then, } ( 1000 + 9 ) ( 1000 - 9 ) = 1009 \times 991 = 999919 \Rightarrow 999919 = ( 1000 + 9 ) ( 1000 - 9 ) = 1009 \times 991. \text { And, the problem said that the number of mice } > \text { the number of cats, so the answer is } 1009 > 991 \rightarrow \boxed { 991 }

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And 1009 , and 991 are all prime factors of 999919 \text { And } 1009 \text { , and } 991 \text { are all prime factors of } \boxed { 999919 }

. . - 3 months, 2 weeks ago
Sean Newell
Jul 5, 2019

We can use 1000^2 = 1 000 000 as a starting point, and then identify that 999 919 is a perfect square (81) less than 1 000 000.

So:

999919 = 1000000 - 81

= 1000^2 - 9^2

= (1000 + 9)(1000 - 9)

= (1009)(991) (which are both prime numbers)

The question states that each cat killed more mice than there were cats, so the answer must be 991

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