Sidebilder
PDF
ePub
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

(2) Change 35.14 from the scale of eight to the scale of

[blocks in formation]

1. If 6, 7, 8, 3, 2 are the digits of a number in the scale of r, beginning from the right, write the number.

2. Find the product of 234 and 125 when r is the base.

3. In what scale will 756 be expressed by 530?

4. In what scale will 540 be the square of 23?

5. In what scale will 212, 1101, 1220 be in arithmetical progression?

6. Multiply 31.24 by 0.31 in the scale of 5.

CHAPTER XXXII.

THEORY OF NUMBERS.

413. Definitions. In the present chapter, by number will be meant positive integer. The terms prime, composite, will be used in the ordinary arithmetical sense.

A multiple of a is a number which contains the factor a, and may be written ma.

An even number, since it contains the factor 2, may be written 2m; an odd number may be written 2m+1, 2m-1, 2m+3, 2m-3, etc.

b

a

A number a is said to divide another number 6 when

is an integer.

414. Resolution into Prime Factors. A number can be resolved into prime factors in only one way.

Let N be the number; suppose N=abc where a, b, c, ..... are prime numbers; suppose also Naßy ..... where α, B, Y, are prime numbers.

Then,

.....

abc... aẞy.....

=

Hence, a must divide the product abc.....; but a, b, c, ...... are all prime numbers; hence a must be equal to some one of them, a suppose.

[blocks in formation]

and so on. Hence, the factors in aẞy..... are equal to those in abc., and the theorem is proved.

415. Divisibility of a Product. I. If a number a divides a product be, and is prime to b, it must divide c.

For, since a divides bc, every prime factor of a must be found in be; but since a is prime to b, no factor of a will be found in b; hence all the prime factors of a are found in c; that is, a divides c.

From this theorem it follows that:

II. If a prime number a divides a product bede...., it must divide some factor of that product; and conversely. III. If a prime number divides b", it must divide b. IV. If a is prime to b and c, it is prime to bc.

V. If a is prime to b, every power of a is prime to every power of b.

416. Theorem. If, a fraction in its lowest terms, is equal

to another fraction, then c and d are equimultiples of a and b.

[merged small][merged small][merged small][merged small][ocr errors]

then = C. Since b will not divide a, it

ad
b

must divide d; hence d is a multiple of b.

Let d=mb, m being an integer; since

[blocks in formation]

Hence, c and d are equimultiples of a and b.

and

From the above theorem, it follows that in the decimal scale of notation a common fraction in its lowest terms will produce a non-terminating decimal if its denominator contains any prime factor except 2 and 5.

α

For a terminating decimal is equivalent to a fraction with a denominator 10". Therefore, a fraction in its lowest terms cannot be equal to such a fraction, unless 10" is a multiple of b. But 10", that is, 2′′ × 5", contains no factors

besides 2 and 5, and hence cannot be a multiple of b, if b contains any factors except these.

417. Square Numbers. If a square number is resolved into its prime factors, the exponent of each factor will be even. For, if N = a2 x ba x ch

[merged small][ocr errors]

Conversely: A number which has the exponents of all its prime factors even will be a perfect square; therefore, to change any number to a perfect square,

Resolve the number into its prime factors, select the factors which have odd exponents, and multiply the given number by the product of these factors.

Thus, to find the least number by which 250 must be multiplied to make it a perfect square.

250 = 2 × 53, in which 2 and 5 are the factors which have odd exponents.

Hence the multiplier required is 2× 5-10.

418. Divisibility of Numbers.

I. If two numbers N and N', when divided by a, have the same remainder, their difference is divisible by a.

For, if N when divided by a have a quotient q and a remainder 2o, then

N=qa+r.

And if N' when divided by a have a quotient q′ and a remainder, then

Therefore,

N'= q'a+r.

N— N'= (q — q') a.

II. If the difference of two numbers N and N' is divisible by a, then N and N' when divided by a will have the same remainder.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

III. If two numbers N and N', when divided by a given number a, have remainders r and r', then NN' and rr' when divided by a will have the same remainder.

[blocks in formation]

Therefore, by II., NN' and 'r' when divided by a will have the same remainder.

As a particular case, 37 and 47 when divided by 7 have remainders 2 and 5 respectively.

Now 37 x 47=1739 and 2× 5 = 10.

The remainder, when each of these two numbers is divided by 7, is 3.

NOTE. From II. it follows that, in the scale of ten :

.....

if the numbers denoted

(1) A number is divisible by 2, 4, 8, by its last digit, last two digits, last three digits, respectively by 2, 4, 8, .....

[ocr errors]

are divisible

(2) A number is divisible by 5, 25, 125, if the numbers denoted by its last digit, last two digits, last three digits, ..... are divisible respectively by 5, 25, 125, .....

(3) If from a number the sum of its digits is subtracted, the remainder will be divisible by 9.

« ForrigeFortsett »