By giving divifors, as the digits nine,
For a true canon I'd give a pint o' wine?

Ladies Diary, 1719.

Divifors I. 2 3.4.5 6

7

8.

3.9.

But as 42, that 6 may be cancelled, being compofed of 2 X 3; 382; and 9 = 3.

... IX 2 X 3X2X5 X 7 X 2 X 7 X 2 X 32520. Q E. F.

2. What particular leaft whole number is that, which being divided by 2, 3, 4, 5, 6, 7, 8, 9, fhall leave a remainder of 1, 2, 3, 4, 5, 6, 7, 8, respectively?

It is plain by the queftion above, that 2520 is the least number that can be divided by nine digits, without a remainder. I 2519, the number re

quired.

... 2520

3. A country girl to town did go,
Some walnuts for to fell,
A gentleman fhe chanc'd to meet,
And thus it her befel:

My pretty maid, fays he to her,
What number have you here?
I can't tell, Sir, fays fhe to him,
But this I'll make appear;

I told them o'er ere I came out,
By fix's, five's, four's, three's, two's,
And, ev'ry time I number'd them,
One remain'd overplus;

I told them o'er by feven's at laft,
And there were no remains ;
If you can find the number out,
Pray take it for your pains.

First, the leaft number that can be divided by 1, 2, 3, 4, 5, 6, without a remainder; viz. 1 X 2X 3X2 X 560. Then 60 +1: 61, will leave 1, when divided by each

number; but 7)61 (8, and 5 remains.

Alfo 60 X 2 +

60 X 3

60 X 4 +

But 60 × 5 +

mits of the conditions of the question.

Then

Then to find the next leaft number which admits of the

But 60 × 12 + 1 = 721, is the next number admitting the conditions aforefaid.

Alfo 721

301420, the common difference of all numbers anfwering the fame conditions.

301, 721, 1141, 1561, 1981, 2401, 2821, &c. ad infinitum, will anfwer the conditions of this question.

4. To find the leaft number of guineas, which being divided by 6, 5, 4, 3 and 2 refpectively, fall leave 5, 4, 3, 2 and 1, refpectively remaining? L. Diary, 1748.

As by the foregoing queftion, I X2 X3 X 2 X 5 = 60, the leaft number, which divided by 1, 2, 3, 4, 5 and 6, leaves no remainder.

·.· 60 — 1 = 59. Q. E. F. as may be eafily proved.

5. Required the leaft number, that being divided by 9, fhall leave for a remainder 6; if divided by 8, the remainder will be 5; if divided by 7, the remainder will be 4; and fo on, each time leaving for a remainder three lefs than the divifor, till, divided by 3, the remainder will be nothing?

As 2520 is the leaft number which can be divided by the nine digits, or by the seven higheft of them, without a remainder,

3 = 2517. Q. E. F. as may be easily

6. Required the three leaft numbers, which divided by 20 fhall leave 19 for a remainder; but, if divided by 19, fhall leave 18, if divided by 18, fhall leave 17; and fo on (always leaving one lefs than the divifor) to unity?

Gentlemen's Diary, 1747.

First, 1, 2, 3, 5, 7, 11, 13, 17, and 19, are prime numbers.

4

Alfo√42, 3√/8 = 2, 1/9 = 3, and √ 16 = 2; And all the rest are compofite numbers.

... 1 × 2 × 3 × 2 × 5 X 7 X 2 × 3 × 11 × 13 X 2 X 17 X 19 = 232792560, the least number that can be divided by

the

the given divifors without a remainder; alfo 232792560 X 2=465585120; and 232792560 x 3 = 698377680, being divided by the given divifors, will leave no remainder.

232792560-1=232792559 the three leaft num4655851201 = 465585119

and 698377680 — 1 — 698377679

-1=698377679

bers. QE. F.

Agreeing with the algebraic process, by Mr. Robinson, in the Gentimen's Diary, 1748.

7. A jolly fine girl did ride on the way,

With plums in a basket, it being market-day;
She rode on but foftly, the weather being hot,
So I ask'd her what number of plums fhe had got:
She faid, the juft number fhe did not well know,
But I'll tell you which way will the true number show.
If you count them by two's, there will then remain one;
If you count them by three's, there refts two when
you've done;

If you count them by four's, the remainder is three;
If by five's, then just four the remainder will be;
If by fix at a time you do count them again,
You'll find, when you've done, that juft five will remain;
But if feven at a time, you do count them o'er all,
The remainder will be then juft nothing at all:
Now what is the number, and to what do they come,
At fourteen a penny, I'd fain know the fum ?

By the third queftion it appears, 60 is the least number that can be divided by the first fix digits, without a remainder, 601 59, the leaft number that can be divided by the faid fix digits, leaving each divifion one less than the divifor; but 59, divided by 7, leaves a remainder.

-

Then 60 X 2 1119, the leaft number that answers the conditions of this queftion.

Alfo 420 is found, by Queftion 3, to be the common difference of numbers, anfwering the fame conditions. . 119, 539, 959, 1379, &c. will admit of the fame conditions. Q. E. F.

. 14) 119 (8 d.

Or 14) 539 (38 = 35. 24 d. }their value.

8. Once old mother Gripe to a market went,
Some butter to fell it was her intent;
At a certain rate per pound fhe it fold,
What she got for it all, as I have been told,

Were

Were two fhillings and two pence farthing juft:
Now how much butter had the old toaft,

And how the might fell her butter per pound,

Is what is required to be found?

Of various anfwers this queftion will admit,

Find them all out, and they will whet thy wit.

=

First, 2 s. 2 d. 105 farthings, which is compofed of thefe odd numbers; viz. I x 3×5x7 105.

This question, and the foregoing, was taken from Tapper's Delight for the Ingenious, for July, Auguft and September, 1711; the folutions my own.

CHAPTER V.

PROGRESSION, VARIATION, COMBINATION, &C.

SECT. I.

ARITHMETICAL PROGRESSION.

ARITHMETICAL PROGRESSION is a rank or feries of num

bers increafing or decreafing by a common difference, or by a continual addition or fubtraction of fome equal numbers.

SI 2
9 8 7 6

3.4.5
5

6.7 .4.3.2

8

Or I.

3.5.7.9.11.13, common difference 2.

Alfo 42.35.28.21.14.7, common difference 7.

In an arithmetical progreffion are five things; any three of which being given, the other two may be found, which admit of twenty different propofitions.

3

J. The

1. The firft term, commonly the leaft

2. The laft term, commonly the greatest extreme. 3. The number of terms.

4. The common excefs, or difference.

5. The aggregate, or fum of all the terms.

We fhall only concern ourselves with fome few of them, but let us premise, that,

I. If any three numbers are in arithmetical progreffion, the fum of the two extremes, viz. the first and last, will be equal to the double of the mean or middle number.

2. If four numbers are in arithmetical progreffion, the fum of the two extremes will be equal to the sum of the

3. Alfo if many numbers be in arithmetic progreffion, the sum of the two extremes will be equal to the fum of any two means that are equally diftant from the extremes.

3

5

7

9. II

13;

viz. 1+13=3+11=5+9=7+7•

4. Every series of numbers in arithmetic progreffion is compofed of the excefs or common difference, so often repeated as there are terms in the progreffion, except the first.

Here the common difference being 3. Then will 2 + 3 = 5·5+3=8.8+3=11. 11+3=1414+3=17, &c.

Hence may be obferved, that the difference between the two extremes (2 and 17) is compofed of the common difference, multiplied into the number of all the terms, except the first.