Some Arithmetical Questions

Excerpts

• Some Arithmetical Questions

  I shall devote the bulk of this chapter to these elementary problems, but I append a few remarks on one or two questions in the theory of numbers.

• Some Arithmetical Questions

  They are given here mainly for their historical---not for their arithmetical---interest; and perhaps a mathematician may well omit them, and pass at once to the latter part of this chapter.

• Some Arithmetical Questions

  I may recall the fundamental rule that no trick, however good, will bear immediate repetition, and that, if it is necessary to appear to repeat it, a different method of obtaining the result should be used.

• Some Arithmetical Questions

  in arithmetic an integral number is denoted by a succession of digits, where each digit represents the product of that digit and a power of ten, and the number is equal to the sum of these products.

• Some Arithmetical Questions

  Then the sum obtained as the result of this last operation will be $1089$.

• Some Arithmetical Questions

  Hence, if $N$ is divided by $a'$, the remainder is $a$.

• Some Arithmetical Questions

  The result is the man’s age in 1906.

• Some Arithmetical Questions

  The reason of the rule is obvious, for he arrives finally at the $(n + 12-m)$th hour from which he started.

• Some Arithmetical Questions

  Problems like this can be worked out only by trial: there are several solutions, of which one is as follows.

• Some Arithmetical Questions

  Obviously, if $A$ calls $43$, then whatever $B$ adds to that, $A$ can win next time.

• Some Arithmetical Questions

  In other words, if the number of counters taken is expressed in the scale of notation whose radix is $n$, then the $(h + 1)$th digit from the right will give the number on the domino selected by $P_h$.

• Some Arithmetical Questions

  The error in each of the foregoing examples is obvious, but the fallacies in the next examples are concealed somewhat better.

• Some Arithmetical Questions

  Now if we put $a = d =1$ and $b = c = -1$ we have four numbers which satisfy the relation $ad = bc$ and such that $a>b$; hence, by the proposition, $c > d$, that is, $-1 > 1$, which is absurd.

• Some Arithmetical Questions

  To the above examples I may add the following questions, which I have often propounded in past years: though not fallacies, they may serve to illustrate the fact that the answer to an arithmetical question is frequently different to what a hasty reader might suppose.

• Some Arithmetical Questions

  The answer is the latter; for in the first year the first clerk receives 100, but the second clerk receives 50 and 55 as his two half-yearly payments and thus receives in all 105.

• Some Arithmetical Questions

  It may be shown that $2^m +1$ is composite if $m$ is not a power of $2$, but of course it does not follow that $2^m + 1$ is a prime if $m$ is a power of $2$.

• Some Arithmetical Questions

  This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.

• Some Arithmetical Questions

  A number is said to be perfect if it is equal to the sum of all its integral subdivisors. Thus the subdivisors of $6$ are $1$, $2$, and $3$; the sum of these is equal to $6$; hence $6$ is a perfect number.

• Some Arithmetical Questions

  Thus only four weights are required, namely, $1$ lb., $3$ lbs., $3^2$ lbs., and $3^3$ lbs.

• Some Arithmetical Questions

  To determine the arrangement of the weights to weigh any given mass we have only to express the number of pounds in it as a number in the ternary scale of notation, except that in finding the successive digits we must make every remainder either $0$, $1$, or $-1$: to effect this a remainder $2$ must be written as $3-1$, that is, the quotient must be increased by unity, in which case the remainder is $-1$.

Equations

• Some Arithmetical Questions

  A = M(a') + 1

  A is a multiple of the product of the other moduli and exceeds a multiple of a' by one.

• Some Arithmetical Questions

  Aa = M(a') + a

  The product Aa leaves remainder a when divided by a'.

• Some Arithmetical Questions

  N = Aa + Bb + Cc + \dotsb

  N is the sum of each remainder multiplied by its chosen coefficient.

• Some Arithmetical Questions

  N-n &= M(a')\,.

  N and the chosen number n differ by a multiple of a'.

• Some Arithmetical Questions

  N &= M(p) + n\,.

  N equals a multiple of p plus the chosen number n, so N leaves remainder n on division by p.

• Some Arithmetical Questions

  x = e

  The hundreds digit of the quotient Q is the tens digit x of the selected number.

• Some Arithmetical Questions

  y=9-r

  The units digit y of the selected number equals nine minus the remainder r.

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  9m-y = a-b + 3(c-d)

  Nine times the integer m minus the units digit y equals a-b plus three times (c-d).

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  (13 - x_1) + (13 - x_2) + \dotsb + (13 - x_p) + r &= 52

  The cards in the p piles (each pile has 13 - x_i cards) plus the r cards left over make up the whole pack of 52 cards.

• Some Arithmetical Questions

  x_1 + x_2 + \dotsb + x_p &= 13p - 52 + r

  The total number of pips on the bottom cards of the p piles equals 13(p - 4) + r.

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  \log(1 + x) = x - \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 - \dotsb

  The logarithm of 1 + x is written as the infinite series x - x^2/2 + x^3/3 - ...

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  \log 1 = 0

  The logarithm of 1 is zero.

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  \sqrt{x-y} = i \sqrt{y-x}

  As an identity in x and y, the square root of x - y equals i times the square root of y - x, where i is either +sqrt(-1) or -sqrt(-1).

• Some Arithmetical Questions

  a + b &= 2c

  If c is the arithmetic mean of a and b, then a + b equals 2c.

• Some Arithmetical Questions

  a:b = c:d

  The ratio a to b is the same as the ratio c to d, so a, b, c, d are in proportion.

• Some Arithmetical Questions

  ad=bc

  If the product of two numbers equals the product of two others, the four numbers are in proportion.

• Some Arithmetical Questions

  \phi=(x^2-y^2)/(x^2 + y^2)^2

  Defines the function phi of x and y used in the integral-order fallacy (footnote to the Fallacies section).

• Some Arithmetical Questions

  \frac{1}{4}\pi =-\frac{1}{4}\pi

  The book states that the iterated integrals of phi in the two orders give pi/4 and -pi/4 respectively, which is the claimed contradiction (footnote; Bertrand).

• Some Arithmetical Questions

  y < n-m

  The number of cards transferred from bottom to top must be less than n - m.

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  m<12

  The starting hour m is less than 12, so n + 12 - m is always positive.

• Some Arithmetical Questions

  m < 20

  The card or domino trick works for any collection of m distinguishable things provided m < 20.

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  (m + n) ! / m! n !

  The number of routes from the top left-hand corner to the bottom right-hand corner of a board of m by n cells, moving only downward or rightward along the ruled lines.

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  (52!)/(13!)^4

  The number of possible distributions of hands at whist from a pack of fifty-two cards, given as a footnote.

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  pr/(ns + r)

  Under the cumulative vote, the least number of supporters who can secure a candidate's election must exceed this quantity.

• Some Arithmetical Questions

  na/(n + 1)

  The man who reaches the greatest distance into the desert occupies this many days before he returns to the starting point.

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  \frac{1}{2}a (1 + \frac{1}{2} + \frac{1}{3}+ \dotsb + 1/n)

  If the explorers may make depots, the longest possible journey occupies this many days.

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  \frac{1}{2}(3^n-1)

  With weights of 1, 3, 3^2, ..., 3^(n-1) pounds one can weigh every integral number of pounds from 1 up to this many pounds, which is the least number of weights for the problem.

• Some Arithmetical Questions

  (1-x^{81})/x^{40} (1-x)

  The sum x^(-40) + x^(-39) + ... + 1 + ... + x^40 is equal to this expression, which can be factored in the way MacMahon used for Bachet's problem.

• Some Arithmetical Questions

  2^p-1

  The number 2^p - 1, which Mersenne's rule tests for primality for various exponents p.

• Some Arithmetical Questions

  2^{p-1} (2^p-1)

  All perfect numbers are believed to be included in this formula, where 2^p - 1 is prime; Euclid proved that every number of this form is perfect and Euler showed it includes all even perfect numbers.

• Some Arithmetical Questions

  2^m + 1

  Fermat asserted that numbers of this form are prime when m = 2^n; the assertion is false, as Euler showed for n = 5.

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  x^n + y^n = z^n

  No integral values of x, y, z satisfy this equation when n is an integer greater than 2.

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  x^2 + 2 = y^3

  Fermat's problem to show that this equation has only one integral solution, namely x = 5, y = 3.

Problems

No exercises in this chapter.