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DIVISIBILITY OF NUMBERS.

161. Numbers exactly divisible by 2; by 2}; by 3}; by 5; by 10.

1. An integral number is exactly divisible by 2 if the right-hand figure is 0, or if the number expressed by its right-hand figure is exactly divisible by 2.

EXPLANATORY NOTĖ. — Every integral number that may be expressed by two or more figures may be regarded as niade up of a certain number of tens and a certain number (0 to 9) primary units, thus: 485 is made up of 48 tens and 5 units ; 4260 is made up of 426 tens and 0 units ; 27562 is made up of 2756 tens and 2 units. But ten is exactly divisible by 2 ; so any number of tens, or any number of tens plus any number of twos, is exactly divisible by 2.

2. Tell which of the following are exactly divisible by 2, and why: 387, 5846, 2750, 2834.

3. Any number, integral or mixed, is exactly divisible by 21, if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 24.

4. Show why the statement made in No. 3 is correct, employing the thought process given in the "Explanatory Note" above.

5. Tell which of the following are exactly divisible by 27, and why: 485, 470, 365, 472), 38473.

6. Any number, integral or mixed, is exactly divisible by 33, if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 33. Show why.

7. Tell which of the following are exactly divisible by 3}, and why: 780, 283}, 5763, 742, 80.

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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 24 contained in 5821 ? * 2. How many times is 2} contained in 375 ? 3. How many times is 24 contained in 4674 ? 4. How many times is 21 contained in 4680 ? 5. How many times is 3} contained in 786% ? † 6. How many times is 3} contained in 543% ? 7. How many times is 3} contained in 8640 ? 8. How many times is 5 contained in 3885 ? I 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 33}; by 12}; 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.

*242 is contained in 5821, (4 X 58) + 1 times. Why? +34 is contained in 7853, (3 X 78) + ?times. Why? 15 is contained in 3885, (2 X 388) + 1 times. Why?

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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 33}, if that part of the number expressed by the figures to the right of the hundreds' figure is exactly divisible by 333.

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 : 364663, 2375, 468333, 38900, 46820.

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

7. Tell which of the following are exactly divisible by 121, and why ; 375, 8374, 6450, 4329, 7467), 3487}, 3425.

8. Any number, integral or mixed, is exactly divisible by 163, if.....

9. Tell which of the following are exactly divisible by 163, and why: 4633}, 5460, 2350, 37400, 275833, 254163.

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.

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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 33} contained in 4666% ?+ 4. How many times is 33} contained in 3433 ? 5. How many times is 12} contained in 4737}?I 6. How many times is 123 contained in 3662} ? 7. How many times is 16} contained in 2533} ? 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 nuniber of nines and 3+2+1 over.

326 is a certain number of nines and 3 + 2 + 6 over; but 3+2+ 6= 11, 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.

166. PROBLEMS. 1. 4625 is a certain number of 9's and

over. 2. 3526 is a certain number of 9's and

Over. 3. 2154 is a certain number of 9's and

over.

* 25 is contained in 2450 (4 X 24) + 2 times. Why?
+ 332 is contained in 46663 (3 X 46) + 2 times. Why?
#12 is contained in 4737; (8 X 47) + 3 times Why?

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 x 3 x 5), is exactly divisible by 2, by 3, by 5, and by (2 x 3), 6, and by (2 x 5), 10, and by (3 x 5), 15. 2. The exact integral divisors of 36, (2 x 2 x 3 x 3), are

and

2, 3,

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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 x 2 x 2 x 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.

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2. Find the g. c. d. of 50, 75, and 125.* OPERATION No. 1.

OPERATION NO. 2. 50 = 2 x 5 x 5.

5150 75 125. 75 3 x 5 x 5.

5 10 15 25. 125 5 x 5 x 5.

2 3 5
5 x 5 = 25, g. c. d.

25, g. c. d.
3. Find the g. c. d. of 80, 100, 140.
4. Find the g. c. d. of 48, 60, 72.

5 x 5

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.

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