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| 91 = 70 + 21 = 7 * 10 + 7 * 3 = 7 * (10+3) = 7 * 13 143 = 130 + 13 = 13 * 10 + 13 = 13 * (10 + 1) = 13 * 11 98 = 70 + 28 = 7 * 10 + 7 * 4 = 7 * (10 + 4) = 7 * 14 1001 = 980 + 21 = 7 * 140 + 7 * 3 = 7 * 143 = 7 * 11 * 13 1001 = 990 + 11 = 11 * 90 + 11 = 11 * (90 + 1) = 11 * 91 = 11 * 13 * 7 1001 = 910 + 91 = 13 * 70 + 13 * 7 = 13 * (70 + 7) = 13 * 77 = 13 * 7 * 11 |
| Potenttial factoring algorithm illustrated: (2 m + 1) * (2 n + 1) = 91 4 * m * n + 2 * m + 2 * n + 1 = 91 4 * m * n + 2 * m + 2 * n = 90 2 * m * n + m + n = 45 45 = 0 mod 3 2 * m * n + m + n = 0 mod 3 m = 0 mod 3 implies n = 0 mod 3. m = 1 mod 3 implies 2*n + 1 + n = 0 mod 3 implies 3*n + 1 = 0 mod 3 implies 1 = 0. m = 2 mod 3 implies 4*n + 2 + n = 0 mod 3 implies 5*n + 2 = 0 mod 3 implies 2*n + 2 = 0 mod 3 implies n+1 = 0 mod 3 implies n = 2 mod 3. First try m = n = 0 mod 3. m = 3*m3; n = 3 * n3 2 * (3 n3) * (3n3) + 3*m3 + 3 * n3 = 45 18 * m3 * n3 + 3*m3 + 3*n3 = 45 6 * m3 * n3 + m3 + n3 = 15 .6*2 + 3 = 15 6 * 1 * 2 + 1 + 2 = 15 m3 = 1; n3 = 2 m = 3*m3'; n = 3 * n3. m = 3*1 = 3; n = 3*2 = 6 (2*m+1) * (2*n+1) = 91 (2*3+1) * (2*6+1) = 91 7 * 13 = 91 |
| Hello Dave. What are some implications of (2 (2 a b + a + b) + 1 ) = 4 a b + 2 a + 2 b + 1 = 2a ( 2b + 1) + (2b+1) = (2a +1) (2b+1) ? |
| 1891 = 30 * 60 + 91 = 30 * 60 + 60 + 31 = (30 + 1) * 60 + 31 = 31 * 60 + 31 = 31 * (60 + 1) = 31 * 61 |
| (a+1)*(b+1) = a*b + (a+b+1) I recall a thing I was told as a kid: Think of a number Double it Add 10 Half it Take away the number you first thought of And the answer is 5 I tried it with so many numbers and was always impressed it was always 5. Then I grew older and learned some algebra. The steps were then: x -> 2x -> 2x + 10 -> x + 5 -> 5 I think the phrase is "well, duh'.  |
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It just a variation on what you had before. In general, if you have four numbers - w, x, y and z - then
(w + x) * (y + z) = w*y + w*z + x*y + x*z
In your case, w = 2a, y = 2b and x and z are both 1, so
(2a + 1) * (2b + 1) = 4ab + 2a + 2b + 1
Ah; ok.
(2 (2 a b + a + b) + 1 )
If a = b = 1, then
(2 (2 a b + a + b) + 1 ) = 9.
If a = 1 and b = 2, then
(2 (2 a b + a + b) + 1 ) = (2 (2 * 2 + 1 + 2) + 1 ) = (2 * (4 + 1 + 2) + 1) = 2 * 7 + 1 = 15.