Background and Topics
This chapter describes techniques related to multiplication, division and repeating decimals, greatest common divisor (GCD), error-checking and divisibility tests, and factoring numbers.
Notes and Errata (If you have any more to contribute, please email me! This is a compendium of feedback) A printer-friendly summary table for all chapters is found here.
Hey, why are the Errata and Notes for Chapter 2 so long?
First, the Notes are good things--they offer additional information. Chapter 2 is by far the most fact-filled chapter of them all, and it makes up over a third of the book. There are many statements per page--it is the most concise chapter of a book that is fairly terse overall. Typos have occurred that do not affect the validity of the methods but are pointed out here. The errors are mostly mistakes in examples that may mislead the reader, not errors in the methods. A few errors were carried over from primary references, and some of the errors and typos were corrected in galley proofs but did not appear in the final layout. I've tried to be as thorough as possible in providing this errata. Finally, this book was written prior to the web, and I did not have the much-appreciated resources of all of you to review the book!
A clarification or elaboration of the text. A simple mistake that does not affect the method presented. An error that may affect a method or the reader interpretation of it.
To get a more intuitive, pictorial feel for the use of Equation 1, as well as other general multiplication/squaring methods, refer to my Mental Calculation Class Handout described and linked in the Additional Materials section of this page. To solve 37x43, Equation 1 can be used to convert this to 40^2 3^2 = 1591. However, if the numbers do not have a midpoint, you can either assume there is one and subtract the difference at the end, or use another shortcut--the two numbers on either side of the midpoint are multiplied, and the product of their offsets is subtracted. So 36x43 = 39x40 3x4 = 1548 (see http://www.urticator.net/essay/5/592.html). "125 x n = 8n,000 / 8 should read 125 x n = n,000 / 8 111 x a||b = a | [(a+b)x11] | b should read 111 x a||b = a | [(a+b)x11] | |b 37 = 111/4 should read 37 = 111/3. I have a funny thing there--for some reason I intuitively think that 111/37 is 4 rather than 3. I don't know why I think that.
58^2 = 3264 should read 58^2 = 3364 A mnemonic scheme (mixed with some mental calculation techniques) for squares and cubes of two-digit numbers can be found in my paper Mnemonics for Squares and Cubes of Two-Digit Numbers found on the webpage for Online Material. Consider it a fun diversion. Technically the period of 1/43 could also be 1, but it obviously it will not be here. are all different and are greater than 5 should read are all different and are greater than 3 as in the example that immediately follows this statement. The factors of 9, 99, and 999 also include 27, and technically also 9, 99, and 999, which are included in the count of 63. Just to be clear about this, corresponding digits in half periods add to 9 for any irreducible fraction s/t with t prime and an even group length, regardless of whether that length is (t-1). For example, this property holds for the primes 11 and 13. (It also holds for some composite denominators with even group lengths, such as 77.) Consider the sentence "In another special case of reciprocal 1/t where t = 2^a x 5^b x t_1^n and t_1 is any prime between 5 and 487, the period of the repeating group equals that of 1/t multiplied by t_1^(n-1)." The reason that the 486-digit recurring group for t = 487 does not double for 1/487^2 is that the recurring group for 1/487 is actually divisible by 487, so when you take 1/487 and divide that by 487, you end up again with a 486-digit recurring group. This is also true for t = 1/3, since 0.3... has a recurring group that is divisible by 3, so 1/3^2 ends up with a single-digit group as well. The next example of this is 56598313, with no further ones known. John McIntosh (see link below) has done an ingenious analysis to show that no other example exists below 2×10^11, and probably not below 5x10^30. "They always will if no factor of t divides 10^n - 1, for n a positive integer." should read "They always will if no factor of t divides 10^n - 1, for n = 1/2 of the group length. "If t is not divisible by 2, 3, or 5 and 1/t has a period of (t-1) digits, this situation will occur as well." This is technically correct, and in fact is stated in this way in the source reference, but in fact it is moot because no non-prime reciprocal 1/t has a recurring period of (t-1). The line that calculates -2^2/43 should have a negative sign inside the parentheses in front of .0465 1/13 = 0.076913 should read 1/13 = 0.076923 "... the adjusted remainder ended up greater than the divisor 80" should read "...the adjusted remainder ended up greater than the divisor 78". "... the dividend by 10n" should read "... the dividend by 10^n" The columns can be seen to be misaligned for the 1/39 calculation. The 3042 at the end of the 1/13 calculation should be 3072. 50/7 = 7.142286 should read 50/7 = 7.142857 In the table for the modified Euclid 's algorithm, 441 = 11x35 + 8x7 could also be represented by 441 = 13x35 - 2x7. 114 in the next-to-last line in Table 1 should be 1140. 57 is not prime, so the combination divisibility test for 47, 53 and 57, while correct as given, is actually a divisibility test for 3, 19, 47 and 53 if we are interested in prime divisibility. Note that if we test for 47x53x59=146969, the product is almost as convenient to work with as it was for 47x53x57=141987. "the prime numbers 3 and 9" should read the numbers 3 and 9" We can add more entries to Table 2: base 80 can have an elevens-test factor of 27, base 100 can have a nines-test factor of 33, and base 1000 can have a nines-test factor of 27 and elevens-test factors of 77 and 91. Omit obviously not greater than N^1/2" at the end of the sentence containing "the largest factor (prime or not) of N. The formula for number of steps in Fermat's factoring method should read (1-k)^2*N^(1/2) / 2k. This is correctly given on page 73. A square ending in 25 can only end in 125, 225, or 625 should read A square ending in 25 can only end in 025, 225, or 625. [(52)^2 - N] should read [(54)^2 - N]. "the next possible value after x=42 would be 114" should read "the next possible value after x=42 would rise further to 138". A case where a/b is approximately 3/2, of course, would benefit from setting k=6, the case a/b approximately 5/3 by letting k=15, and so forth. should read In order to get an integer midpoint of the two factors when a is not an even multiple of b, we either have to multiply by an odd number with factors of about the right ratio, or multiply by a number with two even factors of about the right ratio. A case where a/b is approximately 3/2 would benefit from setting k=24, since 24=6x4 and 6/4=3/2. When a/b is approximately 5/3 (or even 3/2), we can set k=15=5x3, and so forth. The a^5+b^5 factorization should have a plus sign on a^2b^2 term. In Table 4, there are additional values of tx mod m for m=4, 6 and 8, since t(t+1)/2 is not relatively prime to an even m, as shown in this table. The net result is that the cases m=4 and m=8 do not limit possible values of x mod m at all, and are not useful as sieves. The case m=6 retains the same limitations as before, but is redundant to the m=3 sieve, and both of them are inferior to the m=9 sieve. So the triangular modular seives for even m are not useful. "increase the adder" above the N-x^2 equation should read "decrease the adder". All values of r should be calculated to make sure no squares appear (here we should add r2=401 and r3=201). To decrease the number of possible values of x to roughly 521, a mod 8 sieve was also performed. For x=605, we know that (x^2 - N) mod 9 = [(2)^2 - 4] mod 9 = 0; because of the earlier x mod 9 sieve, all of these values of x will give this result. Therefore, only y=234 is possible, since the nines test on the ending 34 implies an initial digit of 2 for the range 0<y<505. An ending of 66 produces no initial digit in this range. As y=234 seems a likely possibility, we cast out elevens to check it again, producing (x^2 - N) mod 11 = [(0)^2 - 7] mod 11 = 5. The y-value 234 is now eliminated because (234)^2 mod 11 = 9; thus x cannot be 605. For x=515, again only y=234 is possible and is eliminated by the elevens test. For x=425, we find y=234 does indeed pass the elevens test. For x=565, we find that y=234 passes the elevens test as well.
should read:
For x=605, y must be less than 505. The mod 9 sieve acting on y-endings of 34 limits these to 234, which also passes the mod 7 sieve. The mod 9 sieve acting on y-endings of 66 in this range limits these to 66 and 366, of which the mod 7 sieve leaves only 66. T he elevens test gives (x^2 - N) mod 11 = [(0)^2 - 7] mod 11 = 4. The y-values 234 and 66 are now eliminated because (234)^2 mod 11 = 9 and (66)^2 mod 11 = 0. Without any possible y, x cannot be 605. For x=515, again y=234 and y=66 are eliminated by the elevens test. For x=425, we find y=234 does indeed pass the elevens test, but 66 does not. For x=565, we find that y=234 passes the elevens test as well, but 66 does not.
72 In the next-to-last equation line, 367 should read 361.
Additional Materials Related to Topics in This Chapter
Developing a "Number Sense" through Mental Multiplication: Mental multiplication is an excellent way to develop an appreciation for the properties and relationships of numbers, sometimes called a number sense. This extensive paper describes simpler methods of multiplying numbers for newcomers to this field that can also be taught to children, as well as more complex methods that may be new even to those with a long interest in this area. A section on squaring numbers, a popular subject, is also included. The paper is available here.
Mental Calculation Class Handout: In 2005 I gave a talk to 4th and 5th graders on mental calculation methods for addition, subtraction and multiplication. I prepared a double-sided handout sheet for the kids to keep as a reference. I later added an additional sheet that would be appropriate for older students who understood algebraic notation. I've attached the three pages here. Although this is meant to one component of a talk, the reference sheets can provide a more readable overview than the book. It shows diagrams in a few cases and includes a number of examples. It is useful generally for adults as well. I hope that readers here will find it personally useful, and I'm happy if teachers want to discuss and distribute it in their classrooms. While the methods are not at all original, the presentation is, and I would appreciate it if it were not generally published, as I am available for talks and this represents some of my material (hence the copyright notice at the top).
A Method for 4x4 Digit Mental Multiplications: A few years ago I wrote a long post to the Yahoo Mental Calculation Group on a convenient method of mentally calculating a 4-digit by 4-digit multiplication. I have polished it into a paper that is available here.
Cited Reference Materials
In the near future, this section will contain relevant sections of some of the references cited at the end of this chapter.
Acknowledgements of Contributors to this Page
A huge thanks goes to John McIntosh, who reviewed this chapter in excruciating detail for me and provided most of the errata and notes you see on this page.
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