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Second highest and lowest numbers
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<blockquote data-quote="Harte" data-source="post: 11062" data-attributes="member: 443"><p><strong>Re: Second highest and lowest numbers</strong></p><p></p><p><div class='quotetop'>QUOTE(\"fanavans\")</div></p><p> </p><p>To understand the fun little game with elevens, remember what multiplication by 11, or 111 or1,111 is saying:</p><p> </p><p>11*12=</p><p> 12</p><p> <u>*11</u></p><p> 12</p><p><u> +120</u></p><p> 132</p><p>Notice the sum in the central area of this. It is only 12+12 with the second 12 offset, of course because 11 is made up of two "1's", the first being 1 times 10, which is the cause of the offset in the sum.</p><p>Because of this, it is easy to remember 11 times anything.</p><p> </p><p>Given a 2 digit number Dd, 11*Dd=D(D+d)d (the parentheses do <strong>not </strong>indicate multiplication here.)</p><p> </p><p>Given a 3 digit number abc, 11*abc = abc+abco, or</p><p> </p><p> abc</p><p><u>+abco</u></p><p>a(a+B)(b+c)c (again,parentheses for clarity,<strong> not</strong> to denote multiplication)</p><p> </p><p>Eleven times anything can be remembered by the fact that the result is just the original number added to it's own string of digits offset by one to the left. Using actual numbers, we can say:</p><p>11*253=2(2+5)(5+3)3=2783. Note the first and last digits remain unchanged after the multiplication.</p><p> </p><p>A similar rule can be observed for 111, only the offset will be one, then two spaces to the left and gets a bit more complicated. In the extreme example provided above by thenumbersix (a good name for a number theory junkie if there ever was one), the fact that all the digits involved are ones makes the rule of the offset in multiplying transparently obvious. The "ones" in the multiplicand make for an easier vision of exactly what multiplying by a string of one's results in.</p><p> </p><p>There is another rule that can help you remember some of your multiplication table. Any multiple of three will, when reduced by the addition of the constituent digits, reduce to a multiple of the number 3. For example, the number 213: 2+1+3=6=multiple of 3 so 213 is a multiple of three. Also, 315765: 3+1+5+7+6+5=27 (multiple of 3) and 2+7=9 (multiple of 3). This can help you check if a number is prime. If the digits add to a multiple of three in this manner, the number cannot be prime.</p><p> </p><p>As a corollary to this "3" business, note that any multiple of the number 9 will reduce to a multiple of 9, then to 9 itself:</p><p>81: 8+1=9</p><p>99: 9+9=18, 1+8=9</p><p>3573: 3+5+7+3=18, 1+8=9</p><p>5895: 5+8+9+5=27, 2+7=9</p><p> </p><p>This is what they called fun in the Mathematics Department.</p><p> </p><p>Harte</p><p>Ps. Edited several times to try to get the vertical products to line up right. H</p></blockquote><p></p>
[QUOTE="Harte, post: 11062, member: 443"] [b]Re: Second highest and lowest numbers[/b] <div class='quotetop'>QUOTE(\"fanavans\")</div> To understand the fun little game with elevens, remember what multiplication by 11, or 111 or1,111 is saying: 11*12= 12 [u]*11[/u] 12 [u] +120[/u] 132 Notice the sum in the central area of this. It is only 12+12 with the second 12 offset, of course because 11 is made up of two "1's", the first being 1 times 10, which is the cause of the offset in the sum. Because of this, it is easy to remember 11 times anything. Given a 2 digit number Dd, 11*Dd=D(D+d)d (the parentheses do [b]not [/b]indicate multiplication here.) Given a 3 digit number abc, 11*abc = abc+abco, or abc [u]+abco[/u] a(a+B)(b+c)c (again,parentheses for clarity,[b] not[/b] to denote multiplication) Eleven times anything can be remembered by the fact that the result is just the original number added to it's own string of digits offset by one to the left. Using actual numbers, we can say: 11*253=2(2+5)(5+3)3=2783. Note the first and last digits remain unchanged after the multiplication. A similar rule can be observed for 111, only the offset will be one, then two spaces to the left and gets a bit more complicated. In the extreme example provided above by thenumbersix (a good name for a number theory junkie if there ever was one), the fact that all the digits involved are ones makes the rule of the offset in multiplying transparently obvious. The "ones" in the multiplicand make for an easier vision of exactly what multiplying by a string of one's results in. There is another rule that can help you remember some of your multiplication table. Any multiple of three will, when reduced by the addition of the constituent digits, reduce to a multiple of the number 3. For example, the number 213: 2+1+3=6=multiple of 3 so 213 is a multiple of three. Also, 315765: 3+1+5+7+6+5=27 (multiple of 3) and 2+7=9 (multiple of 3). This can help you check if a number is prime. If the digits add to a multiple of three in this manner, the number cannot be prime. As a corollary to this "3" business, note that any multiple of the number 9 will reduce to a multiple of 9, then to 9 itself: 81: 8+1=9 99: 9+9=18, 1+8=9 3573: 3+5+7+3=18, 1+8=9 5895: 5+8+9+5=27, 2+7=9 This is what they called fun in the Mathematics Department. Harte Ps. Edited several times to try to get the vertical products to line up right. H [/QUOTE]
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