A Great Greatest Common Divisor Problem

Number Theory Level 3

How many digits does gcd ( 1111...1111 ⏟ $240$ times , 1111...1111 ⏟ $150$ times ) \text{gcd} (\underbrace{1111...1111}_{\text{\$240\$ times}},\underbrace{1111...1111}_{\text{\$150\$ times}}) have?

30 120 60 90

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

X X
Jul 2, 2018

gcd ( 1111...1111 ⏟ $240$ times , 1111...1111 ⏟ $150$ times ) \text{gcd} (\underbrace{1111...1111}_{\text{\$240\$ times}},\underbrace{1111...1111}_{\text{\$150\$ times}})

= gcd ( 1111...1111 ⏟ $240$ times − 1111...1111 ⏟ $150$ times , 1111...1111 ⏟ $150$ times ) =\text{gcd} (\underbrace{1111...1111}_{\text{\$240\$ times}}-\underbrace{1111...1111}_{\text{\$150\$ times}},\underbrace{1111...1111}_{\text{\$150\$ times}})

= gcd ( 1111....1111 ⏟ $90$ times 0000....0000 ⏟ $150$ times , 1111...1111 ⏟ $150$ times ) =\text{gcd}(\underbrace{1111....1111}_{\text{\$90\$ times}}\underbrace{0000....0000}_{\text{\$150\$ times}},\underbrace{1111...1111}_{\text{\$150\$ times}})

= gcd ( 1111....1111 ⏟ $90$ times , 1111...1111 ⏟ $150$ times ) =\text{gcd}(\underbrace{1111....1111}_{\text{\$90\$ times}},\underbrace{1111...1111}_{\text{\$150\$ times}})

= gcd ( 1111....1111 ⏟ $90$ times , 1111...1111 ⏟ $60$ times ) =\text{gcd}(\underbrace{1111....1111}_{\text{\$90\$ times}},\underbrace{1111...1111}_{\text{\$60\$ times}})

= gcd ( 1111....1111 ⏟ $30$ times , 1111...1111 ⏟ $60$ times ) =\text{gcd}(\underbrace{1111....1111}_{\text{\$30\$ times}},\underbrace{1111...1111}_{\text{\$60\$ times}})

= gcd ( 1111....1111 ⏟ $30$ times , 1111...1111 ⏟ $30$ times ) =\text{gcd}(\underbrace{1111....1111}_{\text{\$30\$ times}},\underbrace{1111...1111}_{\text{\$30\$ times}})

= 1111...1111 ⏟ $30$ times =\underbrace{1111...1111}_{\text{\$30\$ times}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...