Fun with Mathematics: July 2024

Tuesday, July 2, 2024

Recreational Mathematics

Find the value of 2¹⁷-1 , 2¹⁸-1 and 2¹⁹+1

When the exponent is even 2¹⁸-1 =( 2⁹)²– 1² = (2⁹-1) (2⁹+1)

2⁹= 512; 2¹⁸-1 = (511) (513) = (500+11) (500+13 ) = 500²+ 500(11+13) + 11x13 = 250000 + 12000 + 143 = 262143

When the exponent is odd

2¹⁷-1= 2 .2¹⁶-2 +1 = 2 (2¹⁶ – 1 ) + 1

= 2[(2⁸-1)(2⁸+1)] +1 = 2[(256-1)(256+1)] +1

= 2(255)(257) +1

= 2[(250+5)(250+7)] +1

=2[250²+250 x 12 +35] +1

= 2[62500 + 3000 + 35]+1

= 2(65535) +1= 131071

2¹⁹+1 = 2.2¹⁸-2 + 3 =2(2¹⁸ – 1) + 3

= 2[(2⁹-1)(2⁹+1)] +3 = 2[(512-1)(512+1)] +3

= 2(511)(513) +3

= 2[(500+11)(500+13)] +3

=2[500²+500 x 24 +143] +3

= 2[250000 +12000 +143]+3

= 2(262143) +1= 524286 + 3= 524289

Fun with Mathematics