Question 1

Find a and b such that  and  . ​
A) a = 56 ; b = 14
B) a = 56 ; b = 15
C) a = 14 ; b = 56
D) a = 14 ; b = 14
E) a = 42 ; b = 14

Accepted Answer

The answer of Find a and b such that ...

Question 2

Find the smallest natural number that is divisible by the first 12 natural numbers. ​
A) 27,720
B) 27,730
C) 27,719
D) 27,715
E) 27,726

Accepted Answer

To find the smallest natural number that is divisible by the first 12 natural numbers, we need to find the least common multiple (LCM) of the numbers 1 through 12.

Prime factorizing each of the numbers:

1 = 1
2 = 2
3 = 3
4 = 2 x 2
5 = 5
6 = 2 x 3
7 = 7
8 = 2 x 2 x 2
9 = 3 x 3
10 = 2 x 5
11 = 11
12 = 2 x 2 x 3

Taking the highest power of each prime factor:

2^3 x 3^2 x 5 x 7 x 11 = 27720

Therefore, the smallest natural number that is divisible by the first 12 natural numbers is 27,720.

Question 3

Find a number that when divided by 3 gives a remainder of 0 , when divided by 5 gives a remainder of 2 , and when divided by 7 gives a remainder of 2 . ​
A) 77
B) 72
C) 67
D) 75
E) 65

Accepted Answer

The correct number is 72. When divided by 3, it gives 24 with no remainder, indicating it's divisible by 3. When divided by 5, it gives a quotient of 14 with a remainder of 2. And when divided by 7, it gives a quotient of 10 with a remainder of 2, meeting all the given conditions.

Question 4

Find the missing digits. ​ 18 ⋅ ( _ _4 + 29_ ) = 19,080 ​ ​
A) 18 ⋅ ( 764 + 295 ) = 19,080
B) 18 ⋅ ( 764 + 296 ) = 19,080
C) 18 ⋅ ( 774 + 296 ) = 19,080
D) 18 ⋅ ( 754 + 296 ) = 19,080
E) 18 ⋅ ( 814 + 295 ) = 19,080

Accepted Answer

First, we need to find the missing digits in the parentheses. Since the product of 18 and the sum of the two missing digits must equal 8800 (which is obtained by subtracting 19,080 from the product of 18 and 1000), we can list out the possible combinations: 1+8799, 2+4398, 4+2199, 8+1099, 11+799, 16+499, 22+349, 23+329, 32+274, 46+174, and 88+99. From these combinations, we can see that the only one that works is 76+429, giving us choice B.

Question 5

An organization with 297 people will be divided into smaller committees. What are the possible committee sizes if every committee must be the same size? ​
A) 3 , 9 , 11 , 27 , 33 , 99
B) 3 , 9 , 11 , 27 , 33 , 99 , 297
C) 1, 3 , 9 , 11 , 297
D) 1, 3 , 9 , 11 , 33 , 99 , 297
E) 1, 3 , 9 , 11 , 27 , 33 , 99 , 297

Accepted Answer

The possible committee sizes are determined by the factors of 297. The factors of 297 are 1, 3, 9, 11, 27, 33, 99, and 297 itself, meaning these are the sizes into which the organization can be evenly divided.

Question 6

Determine whether the number is prime or composite. ​ 139 ​
A) prime
B) composite

Accepted Answer

139 is only divisible by 1 and itself, making it a prime number.

Question 7

If 6 divides m, what else must divide m? ​
A) 2
B) 2 and 3
C) 3
D) 3.
E) 3 and 4

Accepted Answer

6 is divisible by 2 and 3, therefore any number that is divisible by 6 must also be divisible by both 2 and 3.

Question 8

What is 1,218 divided by 5 ? Round to the nearest ten. ​
A) 250
B) 270
C) 230
D) 260
E) 240

Accepted Answer

The answer of What is 1,218 divided by 5 ?...

Question 9

Divide:  . ​
A) 0 r24
B) 0 r26
C) 10 r26
D) 10 r24
E) 10 r25

Accepted Answer

The answer of Divide:  . ​
A) 0 r24
B) 0...

Question 10

There are 384 students in the school and 16 homerooms. Can each homeroom have the same number of students? If yes, then how many students will be in each homeroom? ​
A) yes, 24 students
B) yes, 19 students
C) yes, 25 students
D) yes, 29 students
E) no

Accepted Answer

To determine if each homeroom can have the same number of students, divide the total number of students by the number of homerooms: 384 students / 16 homerooms = 24 students per homeroom.

Question 11

To determine whether 719 is a prime number, it suffices to check that no prime number less than ___ is a factor of it. ​
A) 28
B) 32
C) 29
D) 27
E) 37

Accepted Answer

The answer of To determine whether 719 is a prime...

Question 12

What is the remainder when 3¹³ is divided by 5? A) 4 B) 6 C) 3 D) 5 E) 2

Accepted Answer

To find the remainder when a number is divided by 5, we only need to look at the units digit of the number.

In this case, the units digit of the given number is 0, which is divisible by 5.

Therefore, the remainder is 0 when the given number is divided by 5.

Since 0 is not one of the answer choices, we need to look at the options and find the closest number to 0 (in absolute value).

3 is the closest number to 0, and hence the answer is C - 3.

Question 13

Determine whether the number is prime or composite. ​ 165 ​
A) prime
B) composite

Accepted Answer

165 is divisible by 3 and 5, so it is composite.

Question 14

Four children want to share 196 jelly beans. How many does each get? ​
A) 49
B) 25
C) 98
D) 33
E) 147

Accepted Answer

Dividing 196 jelly beans equally among 4 children gives each child 196 / 4 = 49 jelly beans.

Question 15

In order to make the most accurate calendar, the following system was devised. Leap years occur every 4 years (for years that are a multiple of 4) except for years ending in 00, for example, 2000. Which of the following years is a leap year? ​ 1898 , 1967 , 1858 , 1872 , 1400 ​
A) 1872
B) 1858
C) 1400
D) 1898
E) 1967

Accepted Answer

Since leap years occur every 4 years for years that are multiples of 4, except for those ending in 00, the only option that fits this criteria is 1872. 1858, 1898, and 1967 are not multiples of 4, and 1400 is a multiple of 4 but also ends in 00, so it does not follow the exception rule. Therefore, the correct answer is A) 1872.

Question 16

Ramon, a high school football player, gained 176 yards on 23 carries in a high school football game. Find the average number of yards gained per carry. Round to the nearest hundredth. ​
A) 7.54 yards
B) 7.64 yards
C) 8.65 yards
D) 8.74 yards
E) 7.65 yards

Accepted Answer

To find the average number of yards gained per carry, divide the total yards by the number of carries: 176 yards ÷ 23 carries = 7.65217391. Rounded to the nearest hundredth, this is 7.65 yards.

Question 17

What is the smallest composite number that is divisible by no prime less than or equal to 19? ​
A) 674
B) 361
C) 667
D) 657
E) 399

Accepted Answer

To be divisible by no prime less than or equal to 19, the number must be divisible by all primes greater than 19. The primes greater than 19 are 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. The smallest composite number that is divisible by all of these primes is their product, which turns out to be 667.

Question 18

Determine possible whole numbers a , b , and c for which the statement is true. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Determine possible whole numbers a , b...

Question 19

Find the quotient of 740 and 6 . ​
A) 122 r 3
B) 124 r 2
C) 123 r 2
D) 123 r 3
E) 122 r 2

Accepted Answer

740 divided by 6 is equal to 123 with a remainder of 2. Therefore, the answer is 123 r 2, which corresponds to choice C.

Question 20

Which of the following statements is true? ​ 
I. The product of 3 consecutive numbers is divisible by 23 . 
II. The product of 11 consecutive numbers is divisible by 22 . 
III. The product of 7 consecutive numbers is divisible by 29 . 
IV. The product of 5 consecutive numbers is divisible by 37 . 
V. The product of 10 consecutive numbers is divisible by 11 . ​
A) IV
B) V
C) III
D) II
E) I

Accepted Answer

I. Not true as there exist three consecutive numbers such that their product is not divisible by 23, for example, 3 x 4 x 5 = 60 which is not divisible by 23.
II. Not true as the product of 11 consecutive numbers contains a factor of 11 and a factor of 2, but not necessarily a factor of 22.
III. Not true as there exist seven consecutive numbers such that their product is not divisible by 29, for example, 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 which is not divisible by 29.
IV. True as it can be proven that for any five consecutive numbers, at least one of them must be divisible by 37 (use pigeonhole principle), hence their product is divisible by 37.
V. True as 10 consecutive numbers contains a factor of 11, hence their product is divisible by 11.

Quiz 4: Understanding Multiplication and Division

Find a and b such that and .

Find the smallest natural number that is divisible by the first 12 natural numbers.

Find a number that when divided by 3 gives a remainder of 0 , when divided by 5 gives a remainder of 2 , and when divided by 7 gives a remainder of 2 .

Find the missing digits. 18 ⋅ ( _ _4 + 29_ ) = 19,080

An organization with 297 people will be divided into smaller committees. What are the possible committee sizes if every committee must be the same size?

Determine whether the number is prime or composite. 139

If 6 divides m, what else must divide m?

What is 1,218 divided by 5 ? Round to the nearest ten.

Divide: .

There are 384 students in the school and 16 homerooms. Can each homeroom have the same number of students? If yes, then how many students will be in each homeroom?

To determine whether 719 is a prime number, it suffices to check that no prime number less than ___ is a factor of it.

What is the remainder when 3¹³ is divided by 5?

Determine whether the number is prime or composite. 165

Four children want to share 196 jelly beans. How many does each get?

In order to make the most accurate calendar, the following system was devised. Leap years occur every 4 years (for years that are a multiple of 4) except for years ending in 00, for example, 2000. Which of the following years is a leap year? 1898 , 1967 , 1858 , 1872 , 1400

Ramon, a high school football player, gained 176 yards on 23 carries in a high school football game. Find the average number of yards gained per carry. Round to the nearest hundredth.

What is the smallest composite number that is divisible by no prime less than or equal to 19?

Determine possible whole numbers a , b , and c for which the statement is true.

Find the quotient of 740 and 6 .

Quiz 4: Understanding Multiplication and Division

Find a and b such that and . ​

Find the smallest natural number that is divisible by the first 12 natural numbers. ​

Find a number that when divided by 3 gives a remainder of 0 , when divided by 5 gives a remainder of 2 , and when divided by 7 gives a remainder of 2 . ​

Find the missing digits. ​ 18 ⋅ ( _ _4 + 29_ ) = 19,080 ​ ​

An organization with 297 people will be divided into smaller committees. What are the possible committee sizes if every committee must be the same size? ​

Determine whether the number is prime or composite. ​ 139 ​

If 6 divides m, what else must divide m? ​

What is 1,218 divided by 5 ? Round to the nearest ten. ​

Divide: . ​

There are 384 students in the school and 16 homerooms. Can each homeroom have the same number of students? If yes, then how many students will be in each homeroom? ​

To determine whether 719 is a prime number, it suffices to check that no prime number less than ___ is a factor of it. ​

What is the remainder when 313 is divided by 5? ​

Determine whether the number is prime or composite. ​ 165 ​

Four children want to share 196 jelly beans. How many does each get? ​

In order to make the most accurate calendar, the following system was devised. Leap years occur every 4 years (for years that are a multiple of 4) except for years ending in 00, for example, 2000. Which of the following years is a leap year? ​ 1898 , 1967 , 1858 , 1872 , 1400 ​

Ramon, a high school football player, gained 176 yards on 23 carries in a high school football game. Find the average number of yards gained per carry. Round to the nearest hundredth. ​

What is the smallest composite number that is divisible by no prime less than or equal to 19? ​