Now 40c and 24a are both divisible by 8; therefore b must be divisible by 8. But b is less than 7, it must therefore be zero. And since b is zero, we have 5c=3a, which can only be satisfied when c=3 and a=5.

Thus the number required is 503.

Ex. 6. A number consisting of three digits is doubled by reversing the digits; prove that the same will hold for the number formed by the first and last digits, and also that such a number can be found in only one scale of notation out of every three.

Let the number be abc in the scale of r.

Then we have (abc) × 2= cba.

Since cba is greater than abc, c must be greater than a.

Hence we must have the following equations:

From (i) and (iii) we see that the number represented by ca is double that represented by ac.

Hence, as a is an integer, r 2 must be a multiple of 3, so that the number must be in one of the scales 2, 5, 8, 11, &c., the numbers corresponding to these scales being 011, 143, 275, 3t7, &c.

EXAMPLES XXII.

1. Find the number which has the same two digits when expressed in the scales of 7 and 9.

2. In any given scale write down the greatest and the least number which has a given number of digits.

3. A number of six digits is formed by writing down any three digits and then repeating them in the same order; shew that the number is divisible by 1001.

4. Of the weights 1, 2, 4, 8, &c. lbs., which must be taken to weigh 1027 lbs. ?

5.

Shew that the number represented in any scale by 144 is a square number.

6. Shew that the numbers represented in any scale by 121, 12321 and 1234321 are perfect squares.

7. Find a number of two digits, which are transposed by the addition of 18 to the number, or by converting it into the septenary scale.

8. A number is denoted by 4.440 in the quinary scale, and by 4.54 in a certain other scale. What is the radix of

that other scale?

9. If S be the sum of the digits of a number N, and 2Q be the sum of the digits of 2N, the number being expressed in the ordinary scale, shew that S~Q is a multiple of 9.

10. If a whole number be expressed in a scale whose radix is odd, the sum of the digits will be even if the number be even, and odd if the number be odd.

11. Prove that, in any scale of notation, the difference of the square of any number of three digits and the square of the number formed by reversing the digits is divisible by 2 - 1.

the

12. Prove that, in any scale of notation, the difference of square of any number and the square of the number formed by reversing the digits is divisible by - 1.

13. A number of three digits in the scale of 7, when expressed in the scale of 11 has the same digits in reversed order: find the number.

14. Prove that all the numbers which are expressed in the scales of 5 and 9 by using the same digits, whether in the same order or in a different order, will leave the same remainder when divided by 4.

15. There is a certain number which is expressed by 6 digits in the scale of 3, and by the last three of those digits in the scale of 12. Find the number.

16.

Find a number of four digits in the scale of 8 which when doubled will have the same digits in reverse order.

17. The digits of a number of three digits are in A. P. The number when divided by the sum of its digits gives a quotient 15; and when 396 is added to the number, the sum has the same digits in inverted order. Find the number.

18. Find the digits a, b, c in order that the number 13ab45c may be divisible by 792.

19. Prove that there is only one scale of notation in which the number represented by 1155 is divisible by that represented by 12, and find that scale.

20. Find a number of four digits in the ordinary scale which will have its digits reversed in order by multiplying by 9.

21. In the scale of notation whose radix is r, shew that the number (- 1) (2′′ – 1) when divided by r-1 will give a quotient with the same digits in the reverse order.

22. Shew that, in any scale of notation,

the circulating period consisting of all the figures in order except r-2 which is passed over. For example, in the ordinary scale, 012345679.

=

23. There is a number of six digits such that when the extreme left-hand digit is transposed to the extreme right-hand, the rest being unaltered, the number is increased three-fold. Prove that the left hand digit must be either 1 or 2, and find the number in either case.

24. Find a number of three digits, the last two of which are alike, such that when multiplied by a certain number it still consists of three digits, the first two of which are alike and the same as the former repeated ones, and the third is the same as the multiplier,

S. A.

18

CHAPTER XIX.

PERMUTATIONS AND COMBINATIONS.

236. Definition. The different ways in which r things can be taken from n things, regard being had to the order of selection or arrangement, are called the permutations of the n things r at a time.

Thus two permutations will be different unless they contain the same objects arranged in the same order.

For example, suppose we have four objects, represented by the letters a, b, c, d; the permutations two at a time are ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, and dc.

The number of permutations of n different things taken r at a time is denoted by the symbol „P

n

237. To find the number of permutations of n different things taken r at a time.

Let the different things be represented by the letters a, b, C,......

It is obvious that there are n permutations of the n things when taken one at a time, so that P, = n.

Now, if we take one of the different permutations r — 1 at a time, and place after it any one of the n-(r-1) letters which it does not contain, we shall obtain a permutation of the n things r at a time. We thus obtain n(r-1) different permutations r at a time from every one of the different permutations r-1 at a time.

Since the above relation is true for all values of r, we have in succession,

Multiply all the corresponding members of the above equalities, and cancel all the common factors; we then have

„P, = n (n − 1) (n − 2).............. (n − r + 1).

If all the n things are to be taken, r is equal to n, and we have

„P=n (n - 1) (n-2).......3.2.1.

Definitions. The product n (n − 1) (n − 2) ...2. 1 is denoted by the symbol n or by n! The symbols In and n are read 'factorial n.'

The continued product of the r quantities n, n-1, n-2,......(n-r+1), n not being necessarily an integer in this case, is denoted by n.. Thus nn (n-1) (n-2) (n-3).

Hence we have „P1 = \n, and „P,= ng

n n

n

238. To find the number of permutations of n things taken all together, when the things are not all different.

Let the n things be represented by letters; and suppose p of them to be a's, q of them to be b's, r of them to be c's, and so on. Let P be the required number of per

mutations.