Let a be the number of fourpenny pieces.

Then 4x is their value in pence.

Also 78- is the number of sixpences.

And 6 (78-x) is their value in pence.

Also £1. 16s. 8d. is equivalent to 440 pence.

Hence

4x+6 (78-x)=440,

or 4x+468-6x=440,

from which we find x=14.

Hence there are 14 fourpenny pieces,

and 64 sixpences.

34. A bill of £100 was paid with guineas and half-crowns, and 48 more half-crowns than guineas were used. How many of each were paid?

35. A person paid a bill of £3. 14s. with shillings and half-crowns, and gave 41 pieces of money altogether. How many of each were paid?

36. A man has a sum of money amounting to £11. 13s. 4d., consisting only of shillings and fourpenny pieces. He has in all 300 pieces of money. How many has he of each sort?

37. A bill of £50 is paid with sovereigns and moidores of 27 shillings each, and 3 more sovereigns than moidores are given. How many of each are used?

38. A sum of money amounting to £42. 8s. is made up of shillings and half-crowns, and there are six times as many half-crowns as there are shillings. How many are there of each sort?

39. I have £5. 11s. 3d. in sovereigns, shillings and pence. I have twice as many shillings and three times as many pence as I have sovereigns. How many have I of each sort?

VIII. ON THE METHOD OF FINDING THE

HIGHEST COMMON FACTOR.

119. AN expression is said to be a Factor of another expression, when the latter is divisible by the former.

120. An expression is said to be a Common Factor of two or more other expressions, when each of the latter is divisible by the former.

Thus 3a is a common factor of 12a and 15a,

121. The Highest Common Factor of two or more expressions is the expression of highest dimensions by which each of the former is divisible.

Thus 6a2 is the Highest Common Factor of 12a2 and 18a3, of 10x3y, 15x2y2 and 25x13.

5x2y

NOTE. That which we call the Highest Common Factor is termed by others the Greatest Common Measure or the Highest Common Divisor. Our reasons for rejecting these terms will be given at the end of the chapter.

122. The words Highest Common Factor are abbreviated thus, H. C. F.

123. To take a simple example in Arithmetic, it will readily be admitted that the highest number which will divide 12, 18, and 30 is 6.

Having thus reduced the numbers to their simplest factors, it appears that we may determine the Highest Common Factor in the following way.

Set down the factors of one of the numbers in any order.

Place beneath them the factors of the second number, in such order that factors like any of those of the first number shall stand under those factors.

Do the same for the third number.

Then the number of vertical columns in which the numbers are alike will be the number of factors in the H. C. F., and if we multiply the figures at the head of those columns together the result will be the H. C. F. required.

Thus in the example given above two vertical columns are alike, and therefore there are two factors in the H.C.F.

And the numbers 2 and 3 which stand at the heads of those columns being multiplied together will give the H.C.F. of 12, 18, and 30.

124.

Ex. 1. To find the H.C.F. of a3b2x and a2b3x2.

Ex. 2. To find the H. C. F. of 34a2b6c4 and 51a3b1c2.

34a2b6c4=2 x 17 × aa .bbbbbb.cccc,
51a3b4c2=3x17 × aaa.bbbb .cc;

.. H. C. F. = 17aabbbbcc

5. 18ab2c2d and 36a2bcd2. 6. a3b3, a2b3 and a*b*.

7. 4ab, 10ac and 30bc.

8. 17pq2, 34p2q and 51p3q3.

9. 8x2y3, 12x3y2z3 and 20x1y3z2. 10. 30x1y, 90x2y3 and 120x3yt.

125. The student must be urged to commit to memory the following Table of forms which can or cannot be resolved into factors. Where a blank occurs after the sign it signifies that the form on the left hand cannot be resolved into simpler

=

x3-3x2y + 3xу3 —y3=(x-y)3 ̧ x3 − 3x2+3x−1 = (x−1)3

The left-hand side of the table gives the general forms, the right-hand side the particular cases in which y=1.

126. Ex. To find the H. C.F. of x2-1, x2-2x+1, and x2+2x-3.

127. In large numbers the factors cannot often be determined by inspection, and if we have to find the H.C. F. of two such numbers we have recourse to the following Arithmetical Rule.

"Divide the greater of the two numbers by the less, and the divisor by the remainder, repeating the process until no remainder is left: the last divisor is the H.C. F. required."

Thus, to find the H.C.F. of 689 and 1573.

128. The Arithmetical Rule is founded on the following operation in Algebra, which is called the Proof of the Rule for finding the Highest Common Factor of two expressions.

Let a and b be two expressions, arranged according to descending powers of some common letter, of which a is not of lower dimensions than b.

Let b divide a with p as quotient and remainder c,