50 Fun Facts About Hard Multiplication

Which product of three consecutive primes gives 1001, the trick behind the 7, 11 and 13 divisibility test?

That is why splitting a number into three-digit blocks and alternately adding and subtracting them tests all three primes at once. 1001 is also the first four-digit palindrome.

Multiplying 142857 by which single digit gives 999999?

142857 is the repeating block of 1/7 and the best-known cyclic number: multiplying it by 1 to 6 just rotates its digits.

Using the difference-of-two-squares shortcut, 27 × 33 equals what?

Both numbers sit 3 away from 30, so the product is 30² minus 3², or 900 minus 9. The trick works whenever two numbers have an easily squared average.

Which four-digit number, when multiplied by 9, gives its own digits reversed?

1089 × 9 = 9801. It is also 33 squared and the number that ends the classic 'reverse and subtract, then reverse and add' parlour trick.

Plato praised 5040 as an ideal city population because it is divisible by every number from 1 to 12 except which?

It is also 10 × 9 × 8 × 7, the number of ways to arrange 4 items from 10, and one less than the square 71².

6174 is the fixed point of a digit-sorting routine devised by which Indian mathematician?

Any four-digit number with at least two distinct digits reaches 6174 within seven iterations of sorting the digits, subtracting and repeating.

1729 is the smallest number that is the sum of two positive cubes in two ways. By what name is it known?

Hardy called his taxi's number dull; Ramanujan replied that it was very interesting. It is also a Carmichael number and the first nontrivial taxicab number.

144 is the only nontrivial perfect square that also appears in which famous list of numbers?

Twelve dozen is a gross, and 144 is the twelfth Fibonacci number as well as 12 squared.

4096 can be written as 64², 16³, 8⁴, 4⁶ and 2 to which power?

It is the smallest number with exactly 13 divisors and a superperfect number.

2 to the power 20 equals what?

That is why a mebibyte is 1,048,576 bytes. It is also one of only three powers of two whose digits are all distinct, along with 2⁰ to 2¹⁵ and 2²⁹.

Which is the largest factorial that fits in a 64-bit integer?

12! is the largest that fits in 32 bits. Floating point can hold bigger factorials, but only approximately.

65536 is 2 raised to which power?

It is the smallest number with exactly 17 divisors, and 65536 is the number of code points in a 16-bit character space.

Who coined the term 'repunit' for numbers like 11 and 111 in the 1966 book Recreations in the Theory of Numbers?

The word stands for 'repeated unit'. Repunits are prime for n = 2, 19, 23, 317 and 1031, among a handful of known cases.

Casting out nines checks a multiplication using what single-digit value of each number?

The check works because the remainder of a product must equal the remainder of the product of the remainders. Aryabhata II described it around 950 and it appears in Liber Abaci.

Which Persian polymath, around 1020, described casting out nines as the 'Hindu method' of checking arithmetic?

The earliest surviving description is in the Mahâsiddhânta of the Indian astronomer Aryabhata II, written around 950.

Jakow Trachtenberg developed his rapid mental-calculation system while held where?

The Ukrainian-Jewish engineer used the rules to keep his mind occupied. Doubleday published The Trachtenberg Speed System of Basic Mathematics in English in 1960.

The 1965 book Vedic Mathematics presents its calculation tricks as how many aphorisms and sub-aphorisms?

Scholars note the techniques have practically nothing to do with Vedic-era mathematics; several rely on decimals, which reached India only in the 16th century.

Andrew Booth invented his 1950 multiplication algorithm while researching what at Birkbeck?

Booth's algorithm multiplies two's-complement numbers and remains a staple of computer-architecture courses.

Toom-3 multiplication reduces the nine sub-multiplications of a three-part split to how many?

Andrei Toom introduced the algorithm and Stephen Cook cleaned up its description. Karatsuba is essentially the two-part case.

Karatsuba's 1960 method multiplies two two-digit numbers using how many multiplications instead of four?

Done recursively it beats the schoolbook method's quadratic time and set off decades of research into faster multiplication.

Which Australian computer scientist devised the fast hardware multiplier tree named after him in 1964?

A Wallace tree adds up partial products in parallel using layers of adders, cutting the delay of a hardware multiplier.

Strassen's algorithm multiplies two 2×2 matrices using how many scalar multiplications instead of eight?

The saving compounds recursively for big matrices, at the cost of numerical stability and extra memory for seven auxiliary matrices.

Which 1990 algorithm held the fastest matrix multiplication record until 2010?

The best-known exponent for matrix multiplication now sits below 2.3714, still far from the conjectured 2.

DeepMind's AlphaTensor searched for matrix multiplication algorithms by treating the problem as what?

It built on the reinforcement-learning approach of AlphaZero, and the discovered algorithms were released on GitHub.

Before logarithms in 1614, which trigonometry-based method was the only general way to approximate products quickly?

Astronomers such as Tycho Brahe's circle used product-to-sum identities and sine tables. Its contributors included Wittich, Bürgi, Clavius and Viète.

Genaille–Lucas rulers, presented in 1891, improved on Napier's bones by doing what?

French railway engineer Henri Genaille designed them after Édouard Lucas posed a problem at the Académie in 1885. Mechanical calculators soon made them obsolete.

The Korean finger-counting method chisanbop represents numbers from 0 to what on two hands?

Sung Jin Pai created it in the 1940s and his son Hang Young Pai brought it to the United States in 1977. It supports multiplication and division as well.

On a Japanese soroban, each rod carries how many one-beads below the reckoning bar?

A single five-bead sits above the bar, giving a bi-quinary system in which every rod shows one decimal digit.

The × multiplication sign first appears in a 1618 appendix to whose book on logarithms?

The appendix is attributed to William Oughtred, who used the same symbol in his 1631 Clavis Mathematicae. Unicode encodes it as U+00D7.

Giuseppe Peano's axioms for arithmetic define multiplication with how many axioms?

One handles multiplying by zero, the other by a successor. Associativity and commutativity are then proved from the axioms plus induction.

The rule that a product is zero only if one of the things multiplied is zero is known as what?

It fails in some rings: 2 × 3 is zero modulo 6, so those systems have zero divisors and are not integral domains.

In 1897 August Leopold Crelle published tables giving the product of every pair of numbers up to what?

John Leslie's 1820 quarter-square table had allowed multiplication up to 1000 × 1000 with a few extra steps.

What nickname is given to algorithms, like Harvey–van der Hoeven's, that only pay off for absurdly large inputs?

It builds on the number-theoretic transforms of Schönhage–Strassen but only wins for numbers far larger than anything in practice.

Which two operations does binary 'shift and add' multiplication in software rely on?

By hand the same idea becomes grid or lattice multiplication; the ancient Egyptian doubling method is essentially the decimal version.

Which power of two is the largest with all its digits distinct?

The only such powers are 2⁰ through 2¹⁵, 2²⁰ and 2²⁹.

Which five powers of two are the only known ones whose digits are all even?

Any further example would need an exponent of at least six digits. 512 is one of the first three powers with all-but-last digits odd.

The sum of the first n odd numbers is always what?

1 + 3 + 5 + 7 + 9 = 25 = 5². Galileo's law of odd numbers for falling bodies follows the same pattern.

Which quantity does 5040 also count, being 10 × 9 × 8 × 7?

It is also the 8th superior highly composite number and the 19th highly composite number, and Kahane suggests Plato's use of it was the first appearance of the highly composite idea.

142857 × 3 gives which rotation of its own digits?

Multiples 1 to 6 give the six rotations; times 2 is 285714, times 4 is 571428, times 5 is 714285 and times 6 is 857142.

Whose 1960 conjecture that schoolbook multiplication couldn't be beaten did Karatsuba disprove?

Kolmogorov had stated the conjecture at his Moscow seminar; he was so excited by the refutation that he lectured on it worldwide and published it in 1962 under Karatsuba's name.

Casting out nines relies on a number and its digit sum leaving the same remainder when divided by what?

That is why the digital root of a multiple of nine is always nine and any single-digit error is caught, though swapped digits slip through.

The digital root of 12345 is what?

The digits sum to 15, and 1 + 5 gives 6. Repeating the digit sum until one digit remains is the whole procedure.

Which pair of numbers is a 'Brown number pair', because 7! plus 1 is a perfect square?

5040 + 1 = 5041 = 71². Brocard's problem asks whether any other n beyond 4, 5 and 7 makes n! + 1 square.

The oldest known base-10 multiplication table, on bamboo strips from about 305 BC, comes from which country?

The Warring States-era Tsinghua bamboo slips hold the earliest decimal times table; Babylonian tables were older but base 60.

In many languages the multiplication table is named after which ancient Greek mathematician?

French, Italian and Russian call it the Table of Pythagoras; Nichomachus later included one in his Introduction to Arithmetic.

How many single-sided Napier's rods are needed to multiply four-digit numbers with repeated digits?

Four copies of the table for each digit 0–9 are needed, so 40 rods; square rods let all 40 tables fit on 10.

What name did John Napier invent for calculation with his rods, based on lattice multiplication?

Rabdology, from the Greek for 'rod', reduces multiplication to addition and division to subtraction.

Which French mathematician introduced the exclamation-mark notation for factorials in 1808?

Kramp's n! notation won out over many alternatives; Stirling's approximation for large factorials dates from 1729.

The Hebrew mystical text Sefer Yetzirah lists factorials up to which value while counting possible words?

The Talmudic-era book of creation lists factorials up to 7! to explore how many words the Hebrew alphabet can form.

Which Greco-Roman mathematician noted the fourth perfect number, 8128, around AD 100?

Nicomachus knew the first four perfect numbers and wrongly claimed they end alternately in 6 and 8.