Chapter 6

Divided We Stand

Division is the most interesting and complex of the Big Four operations (addition, subtraction, multiplication, and division). When you divide two numbers, you divide the dividend by the divisor, and the result is the quotient. For example:

Dividend		Divisor		Quotient
24		8	=	3

Integer division — that is, division with whole numbers only — always results in a remainder (which may be 0).

Dividend		Divisor		Quotient	Remainder
26		8	=	3	r2

The remainder is a whole number from 0 to one less than the number you’re dividing by. For example, when you divide any number by 8, the remainder must be a whole number from 0 to 7, inclusive.

The Problems You’ll Work On

This chapter focuses on the following concepts and skills:

Understanding integer division — that is, division with a remainder

Knowing some quick rules for divisibility

Finding the remainder to a division problem without dividing

Distinguishing prime numbers from composite numbers

What to Watch Out For

Following are some rules and tips to utilize when working division problems:

One integer is divisible by another when the result of division is a remainder of 0. For example, , so 54 is divisible by 3.

A prime number is divisible by exactly two numbers: 1 and the number itself. For example, 17 is a prime number because it’s divisible only by 1 and 17.

A composite number is divisible by 3 or more numbers. For example, 25 is a composite number because it’s divisible by 1, 5, and 25.

The number 1 is the only number that is neither prime nor composite.

Determining Divisibility

181–200

181. Which of the following numbers are divisible by 2?

i. 32

ii. 70

iii. 109

iv. 8,645

v. 231,996

182. Which of the following numbers are divisible by 3?

i. 51

ii. 77

iii. 138

iv. 1,998

v. 100,111

183. Which of the following numbers are divisible by 4?

i. 57

ii. 552

iii. 904

iv. 12,332

v. 7,435,830

184. Which of the following numbers are divisible by 5?

i. 190

ii. 723

iii. 1,005

iv. 252,525

v. 505,009

185. Which of the following numbers are divisible by 6?

i. 61

ii. 88

iii. 372

iv. 8,004

v. 1,001,010

186. Which of the following numbers are divisible by 8?

i. 881

ii. 1,914

iii. 39,888

iv. 711,124

v. 43,729,408

187. Which of the following numbers are divisible by 9?

i. 98

iii. 324

iii. 6,009

iv. 54,321

v. 993,996

188. Which of the following numbers are divisible by 10?

i. 340

ii. 8,245

iii. 54,002

iv. 600,010

v. 1,010,100

189. Which of the following numbers are divisible by 11?

i. 134

ii. 209

iii. 681

iv. 1,925

v. 81,928

190. Which of the following numbers are divisible by 12?

i. 81

ii. 132

iii. 616

iv. 123,456

v. 12,345,678

191. What’s the greatest power of 10 that’s a factor of 87,000?

192. What’s the greatest power of 10 that’s a factor of 9,200,000?

193. What’s the greatest power of 10 that’s a factor of 30,940,050?

194. The number 78 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one answer is possible.)

195. The number 128 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one answer is possible.)

196. The number 380 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one answer is possible.)

197. The number 6,915 is divisible by which numbers from 2 to 6, inclusive? (Note: More than one answer is possible.)

198. You know that the number 56 is divisible by 7 (because .) Using that information, what is the remainder when you divide ?

199. You know that the number 612 is divisible by 9 (because 6 + 1 + 2 = 9). Using that information, what is the remainder when you divide ?

200. You know that the number 9,000 is divisible by 6 (because it’s an even number whose digits add up to 9, which is divisible by 3). Using that information, what is the remainder when you divide ?

Working with Prime and Composite Numbers

201–210

201. Which of the following are prime numbers and which are composite numbers?

i. 39

ii. 41

iii. 57

iv. 73

v. 91

202. Is 143 a prime number?

203. Is 151 a prime number?

204. Is 161 a prime number?

205. Is 223 a prime number?

206. Is 267 a prime number?

207. Which two different prime numbers is 93 divisible by?

208. Which two different prime numbers is 297 divisible by?

209. Which two different prime numbers is 448 divisible by?

210. Which three prime numbers is 293,425 divisible by?