An 8th-grader number theory problem

Xiangdang

An 8th-grader math contest problem:

 

Prove that the unit-digits of a Mersenne Prime(for p >2, 2^p-1) will be either 1 or 7.

example: 

p=17, 2^p-1=131,071

p = 31, 2^p-1=2,147,483,647

 

p.s. So far there are only 46 confirmed Mersenne primes. (Wikipedia)

www.ddhw.com

 

idiot94

不要被那个深奥的梅森素数吓着了，其实和它没有什么太大关系的。 www.ddhw.com

yma16

Consider 2^n where n=3,4,5... The last digits of the sequence are 8,6,2,4,8,6,2,4,... The 4 numbers repeat. So 2^p-1 cannot have 9 as the last digit. In order to get 3, 2^p must be 4. But when 2^n has 4 as the last digit, n is always even.www.ddhw.com