Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 61
the result. Or a quite arbitrary sequence of operations may
be used, ending with mUltiplication by some multiple of 9.
7. How TO GUESS AN UNKNOWN NUMBER FROM ITS RE­
MAINDERS WHEN DIVIDED BY A SET OF GIVEN NUMBERS.
Suppose we use the divisors 4, 5, and 7. Let a, b, and c
be the remainders after division by 4, 5, and 7 respectively.
Then
N == 105a + 56b + 120c (modulus = 4·5·7 = 140).
In order that the result be unique the number to be guessed
should be required to be less than 140.
Note that the coefficient of a, 105, in the expression for N
has been so chosen that it is exactly divisible by 5 and 7, and
gives the remainder 1 when divided by 4. Similarly,56 == 0
(m = 4 and m = 7); 56 == 1 (m = 5); and 120 == 0 (m = 4
and m = 5); 120 == 1 (m = 7).
If the divisors are 3, 5, and 7, and the respective remainders
a, band c, the selected number will be
N == 70a+ 21b+ 15c
When the divisors are 7, 11, and 13,
N == 715a + 364b + 924c
(m = 105).
(m = 1,001).
More generally, if p, q, r, ... are prime each to each, and
a, b, c, .•. are the respective remainders from these divisors,
then nwnbers P, Q. R, ... can be found so that
N == Pa + Qb + Rc + ... (m = pqr . .. ).
To find P, form the product qr . •. of all the divisors except p,
and select the lowest multiple of this product which leaves
the remainder 1 on division by p. Similarly for Q, R, and so
on. The number to be selected should be kept less than the
product of all the divisors.
8. How TO GUESS A NUMBER LESS THAN 1,000. Ask the