Special Numbers

Question 1

Say -3 - 2 using addition

Answer 1

negative 3 plus negative 1

Question 2

Adding and subtracting are inverse operations. True or False

Answer 2

True

Question 3

On a number line what will you do to add a positive number?

Answer 3

To add a positive number move right on the number line

Question 4

On a number line what will you do to subtract a positive number

Answer 4

To subtract a positive number move left on the number line

Question 5

On a number line what you do to add a negative number?

Answer 5

To add a negative number move left on number line

Question 6

On a number line what will you do to subtract a negative number?

Answer 6

To subtract a negative number move right on number line

Question 7

What is a magic square?

Answer 7

A magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an nxnmatrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n(n²+1)/2.

Question 8

Multiplication and Division are inverse operations. True or False?

Answer 8

True

Inverse operation means one undoes the other.

Example: -6 / -3 = 2, -3 x 2 = -6

Question 9

To check divisibility of larger numbers we can check divisibility by factors. True or False?

Answer 9

True

• Example-1: To check if 345 is divisible by 15, we can check if 345 is divisible by 3 and 5. Since 3 x 5 = 15 345 is divisible by 15.
• Example-2: To check if 1362 is divisible by 18, check for divisibility by 2 and 9. 1362 is divisible by 2 but not by 9, hence 1362 is not divisible by 9.

Question 10

What are common factors?

Answer 10

Factors are the numbers you multiply together to get another number

When you find the factors of two or more numbers, and then find some factors are the same (“common”), then they are the “common factors”.

Example: 12 and 30
• The factors of 12 are: 1, 2, 3, 4, 6 and 12
• The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30
So the common factors of 12 and 30 are: 1, 2, 3 and 6

Question 11

Divisibility using factors

Pranay said 410 is divisible by 20 because 410 is divisible by 2 and 10

Priya said the opposite because 410 is divisible by 5 but not 4

Who is correct?

Answer 11

Pranay used 2 x 10 = 20

Priya used 4 x 5 = 20

LCM of 2 and 10 is 10

LCM of 4 and 5 is 20

Hence Priya is correct.

Question 12

To check 258 is divisible by 12

Pranay used 3 and 4 factors

Priya used 2 and 6

Who numbers are right?

Answer 12

Factors of 258 are 2, 3, 6

Factors of 12 are 2, 3, 6

Hence common factors which make up 12 are 2 x 6

Priya is correct this time

Question 13

Sum of any consecutive five numbers are divisible by 5.

True of False?

Answer 13

True

Proof: Sum of 5 consecurituve numbers starting with n are

n + (n+1) + (n+2) + (n + 3) + (n + 4) =

5n + 10 = 5 x (n+2)

Hence sum of any consecutive five numbers are divisible by 5.

Question 14

Let n be odd. Sum of any consecutive n numbers is divisible by n.

True of False?

Answer 14

True

Proof: Sum of n consecutive numbers starting with x (any integer) can be written as

x + (x + 1) + (x + 2) + (x +3) + … + (x + n) =

nx + (1+ 2 + 3 +4 + .. + n) =

nx + n(n+1)/2 =

n (x + (n+ 1) / 2)

Since n is odd n+1 is even and hence divisible by 2. Thus x + (n+1) /2 is an integer. Hence proved.

Question 15

Largest factor of a given integer n is n/2

True or False?

Answer 15

True

Proof: Say n = a x b

• if a has be the largest number then b has to be smallest number.
• b cannot be 1 because a then will be = n
• hence smallest number b can take is 2. Hence a = n / 2

Question 16

To test if a number is prime, try to divide it by each of the prime numbers < n^1/2 in turn

True or False?

Answer 16

True

Proof: n = a x b

• if a < n^1/2 then b > n^{1/2 and vice versa}
• hence factors after n^1/2 repeats themselves
• All even factors can be eliminated since if n is divisible by even number then its divisible by 2

More elaborate and complete proof is here

Question 17

To test if 149 is prime, it is sufficient to try dividing it by 2, 3, 5, 7, and 11

True or False?

Answer 17

True

149^1/2 ~= 12 (since 12 x 12 = 144)

Prime numbers less than 12 are 2,3,5,7 and 11

Question 18

Two prime numbers are added. The answer is 36. What could the numbers be?

Find all possible answers.

Answer 18

You can write 36 as

1+35

2+34

3+33

4+32

…

16 + 20

17 + 19

18 + 18

19 + 17

20 + 16

….

33 + 3

34 + 2

35 + 1

Things repeat after 18 + 18

So it is sufficnet to check till N/2

Question 19

Every even integer greater than 2 can be expressed as the sum of two primes.

True of False?

Answer 19

True

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 7 + 5

…

100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

…

This is callled Goldbach’s conjecture

Question 20

Every even number greater than 4 is sum of two prime numbers.

True or False?

Answer 20

True

6 = 3 + 3

8 = 3 + 5

10 = 5 + 5

Question 21

Every odd number greater than 7 is sum of three prime numbers

True or False?

Answer 21

True

Examples:

11 = 2+2+7

13 = 3 + 3 + 7

Question 22

What is a prime factor?

Answer 22

A prime factor is factor that is prime

Example: Factors of 12 are 1,2,3,4,6 and 12

Of these, 2 and 3 are prime factors.

Question 23

What is Highest Common Factor (HCF)?

Answer 23

The HCF of two numbers is the largest factor common to both.

Example:

The factors of 18 are 1,2,3,6,9, 18

The factors of 30 are 1,2,3,5,6,10,15,30

The HCF of 18 and 30 is 6

Question 24

What is Lowest Common Multiple (LCM) ?

Answer 24

The LCM of two numbers is the smallest number that is a multiple of both.

Example:

The multiples of 9 are 9,18,27,36,45,54,63,…

The multiples of 15 are 15, 30, 45, …

The LCM of 9 and 15 is 45

Question 25

Squaring and Square root are inverse operations

True or False

Answer 25

True

3 → Square → 9

9 →Square Root → 3