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Xiangdang 发表于 2009-3-28 02:20:23

An 8th-grader number theory problem

<br /="/"/><div>An 8th-grader math contest problem:</div><div> </div><div><u>Prove</u> that the unit-digits of a Mersenne Prime(for p >2, 2^p-1) will be either 1 or 7.</div><div>example:  </div><div>p=17, 2^p-1=131,07<u><strong>1</strong></u></div><div>p = 31, 2^p-1=2,147,483,64<strong><u>7</u></strong></div><div><strong><u></u></strong> </div><div><u><strong>p.s.</strong></u> So far there are only 46 confirmed Mersenne primes. (Wikipedia)</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div>

idiot94 发表于 2009-4-1 23:50:06

为什么没有人做了呢？这个题目比下面那个六年级的容易很多啊。。

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>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。</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

yma16 发表于 2009-4-8 04:19:00

回复：An 8th-grader number theory problem

<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/>Consider 2^n where n=3,4,5...<br /="/"/><br /="/"/>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.<span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>

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