2 - A Calculator

Number Theory Level 2

Ana Maria pressed the keys 1 , 9 , × , 1 , 0 , 6 1, 9, \times, 1, 0, 6 and = = in one calculator and the result in 2014 appeared on the display. The highest digit that she pressed was 9.

Then she cleared the display and pressed six keys (instead of seven as above) to obtain 2014 again. In this second multiplication, which was the highest number whose key she have pressed?

6 7 8 9

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Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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3 solutions

Maxwell Johnson
Jun 17, 2014

106/2 = 53

19 x 2 = 38

53 x 38 = 2014

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Athira Anilkumar
Jun 19, 2014

19 x 106= 2014 (given). since only four number keys have been pressed, it has to be the product of two 2-digit numbers. so, 106/2=53

19x2=38

53x38=2014

1 Helpful 0 Interesting 0 Brilliant 0 Confused

The divisors of 2014 are:

1 | 2 | 19 | 38 | 53 | 106 | 1007 | 2014

Have only one solution:

38 × 53 = 2014 38 \times 53 = 2014

You press in order these keys: 3, 8, X, 5, 3 and =. Six keys. The largest number that she was pressed is 8.

=

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Very good.

Rafael Tages Melo - 6 years, 12 months ago

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