The Taxi Driver

Number Theory Level 1

Sometimes taxi drivers can be very rude. One taxi driver and I had an occasion travel with was lacking in courtesy, and so I asked his number. The driver gave me a cynical smile and said, well of you divide this number by 2,3,4,5 and 6 you will find that there is always remainder 1. But if you divide it with 11, there is no remainder. He also said there is no taxi in the town having less number than him. What is the number.

60 120 121 61

3 solutions

Daniel Liu
Jul 1, 2014

Note that none of the answer choices are divisible by $11$ except $121$ . Thus, the answer is $\boxed{121}$ .

Next time, consider using the answer blank instead of multiple choice.

Sunil Pradhan
Jul 9, 2014

If 2 options were divisible by 11 then what to do ?

No. ÷ 2, 3, 4, 5, 6 every time remainder is 1

No. ÷ 11 remainder is 0

First find the number which satisfy the 1st condition.

Such least number is (LCM of divisors ) + remainder

LCM of 2, 3, 4, 5, 6 is = 60

so least number is 60 + 1 = 61

to find number satisfying 2nd condition find multiple of 60 and add 1 till you get the proper number

= 60 × 2 + 1 = 121

The problem is equivalent to: Find least nonnegative $11k$ if

$11k\equiv 1\equiv 121\pmod{\!60}\stackrel{:11}\iff k\equiv 11\pmod{\!60}$

$\,\Rightarrow\, k=60m+11$ . So $11k=11(60m+11)=660m+\boxed{121}$ .

mathh mathh - 6 years ago

Rohit Joseph
Jan 13, 2016

have a look on the options,we see that all the conditions are false except for 121

The Taxi Driver

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