Divisibility of Numbers.

8. Any integral number is exactly divisible by 5, if its right-hand figure is 0 or 5. Show why and give examples. 9. Any integral number is exactly divisible by 10 if its right-hand figure is

162. PROBLEMS.

1. How many times is 2 2. How many times is 2 3. How many times is 2 4. How many times is 2

5. How many times is 33 6. How many times is 3 7. How many times is 3

contained in 582 ?*
contained in 375?
contained in 4671⁄2 ?
contained in 4680?
contained in 7863 ? †
contained in 543§ ?
contained in 8640?

8. How many times is 5 contained in 3885 ? ‡ 9. How many times is 5 contained in 1260 ? 10. How many times is 2 contained in 8646 ?

163. Numbers exactly divisible by 25; by 331; by 121; by 163; by 20; by 50.

1. Any integral number is exactly divisible by 25 if its two right-hand figures are zeros or if the part of the number expressed by its two right-hand figures is exactly divisible by 25.

EXPLANATORY NOTE —Every integral number expressed by three or more figures may be regarded as made up of a certain number of hundreds and a certain number (0 to 99) primary units; thus 4624 is made up of 46 hundreds and 24 units; 38425 is made up of 384 hundreds and 25 units; 8400 is made up of 84 hundreds and 0 units. But a hundred is exactly divisible by 25; so any number of hundreds, or any number of hundreds plus any number of 25's is exactly divi. sible by 25.

Divisibility of Numbers.

2. Tell which of the following are exactly divisible by 25, and why: 37625, 34836, 27950, 38575.

3. Every number, integral or mixed, is exactly divisible by 333, if that part of the number expressed by the figures to the right of the hundreds' figure is exactly divisible by 33.

4. Show why the statement made in No. 3 is correct, employing the thought process given in the "Explanatory Note" under No. 1 on the preceding page.

5. Tell which of the following are exactly divisible by 33, and why: 36466%, 2375, 468331, 38900, 46820.

6. Any number, integral or mixed, is exactly divisible by 121, if the part of the number expressed by the figures to the right of the hundreds' figure, is exactly divisible by 12. Show why.

7. Tell which of the following are exactly divisible by 12, and why: 375, 8371, 6450, 4329, 7467, 3487, 3425.

8. Any number, integral or mixed, is exactly divisible by 16, if......

9. Tell which of the following are exactly divisible by 16, and why: 46331, 5460, 2350, 37400, 27583, 25416.

10. Any integral number is exactly divisible by 20 if the number expressed by its two right-hand figures is exactly divisible by 20. Show why.

11. Tell which of the following are exactly divisible by 20, and why: 3740, 2650, 3860, 29480, 3470.

12. Tell which of the following are exactly divisible by 50, and why: 2460, 3450, 6800, 27380, 25450.

Divisibility of Numbers.

164. PROBLEMS.

1. How many times is 25 contained in 2450?*
2. How many times is 25 contained in 3775?
3. How many times is 333
4. How many times is 33
5. How many times is 12
6. How many times is 123

contained in 4666 ?† contained in 3433 ? contained in 47371 ? ‡ contained in 3662? contained in 2533?

7. How many times is 16
8. How many times is 163 contained in 4550?

165. Numbers exactly divisible by 9.

1. Any number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9.

EXPLANATORY NOTE.—Any number more than nine is a certain number of nines and as many over as the number indicated by the sum of its digits. Thus, 20 is two nines and 2 over; 41 is four nines and 4 + 1 over; 42 is four nines and 4+ 2 over; 200 is twenty-two nines and 2 over; 300 is thirty-three nines and 3 over; 320 is a certain number of nines and 3+2 over; 321 is a certain number of nines and 3+2+1 over.

326 is a certain number of nines and 3 + 2 + 6 over; but 3+2+ 611, or another nine and 2 over.

2. Read the "Explanatory Note" carefully, and tell which of the following are exactly divisible by 9: 3256, 4266, 2314, 2574.

Divisibility of Numbers.

167. PRIME FACTORS AND EXACT DIVISORS.

1. Any integral number is exactly divisible by each of its prime factors and by the product of any two or more of its prime factors. Thus, 30, (2 × 3 × 5), is exactly divisible by 2, by 3, by 5, and by (2 × 3), 6, and by (2 × 5), 10, and by (3 × 5), 15.

2. The exact integral divisors of 36, (2 × 2 × 3 × 3), are 2, 3,

and

168. PRIME FACTORS, COMMON DIVISORS, AND GREATEST COMMON DIVISORS.

1. Any prime factor or any product of two or more prime factors common to two or more numbers is a common divisor of the numbers. Thus, the numbers 30, (2 x 3 x 5), and 40, (2 × 2 × 2 × 5), have the factors 2 and 5 in common. So the common divisors of 30 and 40 are 2, 5, and 10, and the greatest common divisor is 10.

RULE. To find the greatest common divisor of two or more numbers, find the product of the prime factors common to the numbers.

2. Find the g. c. d. of 50, 75, and 125.*

3. Find the g. c. d. of 80, 100, 140.

4. Find the g. c. d. of 48, 60, 72.

5. Find the g. c. d. of 64, 96, 256.

The letters g. c. d. are sometimes used instead of the words greatest common divisor.

Divisibility of Numbers.

6. Find the g. c. d. of 640 and 760.

Operation.

640)760(1

640

120)640(5

600
40)120(3
120

Explanation.

The number 760 is an integral number of times the g. c. d., whatever that may be; so is the number 640. We make an incomplete division of 760 by 640 and have as a remainder the number 120. Since 640 and 760 are each an integral number of times the g. c. d., their difference, 120, must be an integral number of times the g. c. d.: for, taking an integral number of times a thing from an integral number of times a thing, must leave an integral number of times the thing. Therefore, no number greater than 120 can be the g.c.d. But if 120 is an exact divisor of 640, it is also an exact divisor of 760, for it will be contained one more time in 760 than in 640. We make the trial and find that 120 is not an exact divisor of 640; there is a remainder of 40. Since 600, (120 × 5), and 640 are each an integral number of times the g. c. d., 40 must be an integral number of times the g. c. d. But if 40 is an exact divisor of 120 it is an exact divisor of 600, (120 × 5), and 640, (40 more than 600), and 760, (120 more than 640). We make the trial and find that it is an exact divisor of 12), and is therefore the g. c. d. of 640 and 760.

From the foregoing learn that any number that is an exact divisor of two numbers is an exact divisor of their difference.

169. From the foregoing make a rule for finding the g. c. d. of two numbers and apply it to the following

Find the g. c. d.

PROBLEMS.

1. Of 380 and 240.
2. Of 275 and 155.

3. Of 144 and 96.
4. Of 1728 and 288.

5. Of 650 and 175.

6. Of 540 and 450.

7. Of 320 and 860.

8.

Of 475 and 350. 9. Of 390 and 520.

10. Of 450 and 600.

(a) Find the sum of the ten results.