Quiz-1

n divided by 5 yields a remainder equal to 3 is written as follows 
n = 5 k + 3 , where k is an integer.
add 2 to both sides of the above equation to obtain 
n + 2 = 5 k + 5 = 5(k + 1)
The above suggests that n + 2 divided by 5 yields a remainder equal to zero. 

Video solution:

5 and 12 it must a multiple of the lcm of 3, 5 and 12 which is 60. 
n = 60 k
n + 60 is also divisible by 60 since 
n + 60 = 60 k + 60 = 60(k + 1)

Video solution:

It is the lcm of 5, 7 and 20 which is 140. 

When n is divided by 8, the remainder is 3 may be written as 
n = 8 k + 3
multiply all terms by 6 
6 n = 6(8 k + 3) = 8(6k) + 18
Write 18 as 16 + 2 since 16 = 8 * 2. 
= 8(6k) + 16 + 2
Factor 8 out. 
= 8(6k + 2) + 2
The above indicates that if 6n is divided by 8, the remainder is 2.

We first expand (2n + 2)² 
(2n + 2)² = 4n ² + 8 n + 4
Factor 4 out. 
= 4(n ² + 2n + 1)
(2n + 2)² is divisible by 4 and the remainder is equal to 0.

 

Let xy be the whole number with x and y the two digits that make up the number. The number is divisible by 3 may be written as follows 
10 x + y = 3 k
The sum of x and y is equal to 9. 
x + y = 9
Solve the above equation for y 
y = 9 - x Substitute y = 9 - x in the equation 10 x + y = 3 k to obtain.
10 x + 9 - x = 3 k

Solve for x 

x = (k - 3) / 3

x is a positive integer smaller than 10 

Let k = 1, 2, 3, ... and select the first value that gives x as an integer. k = 6 gives x = 1 

Find y using the equation y = 9 - x = 8 

The number we are looking for is 18 .

 It is divisible by 3 and the sum of its digits is equal to 9 and it is the smallest and positive whole number with such properties.

Video solution:

we can also test for divisibilty by adding the digits and if the result is divisible by3 then the number is divisible by 3. 
3 + 3 + 9 = 15 , divisible by 3.
3 + 4 + 2 = 9 , divisible by 3.
5 + 5 + 2 = 12 , divisible by 3.
1 + 1 + 1 + 1 = 4 , not divisible by 3.
The number 1111 is not divisible by 3

Video solution:

To divisibility rule of 12 is that a number should be divisible by 4 and 3.
4 – The number is divisible by 4 as its last two digits end with 12 and 12 is the multiple of 4.
3 – To make a number divisible by 3 we need to add all its digits and the sum should be a multiple of 3.
2 + 9 + 6 + * + 1 + 2 = 20 + *
To make the sum a multiple of 3 we need to add 1 to it.So, 20  1 = 21 and hence 1 is the smallest number that should be placed instead of * to make it divisible by 12.
Therefore, the answer is Option 1: 1.

Step 1: 224, the numbers we get after separation are 22 and 4
Step 2: Multiply the unit digit with 2
2 x 4 = 8
Step 3: subtract the number from the rest of the number.
22 – 8 = 14
Step 4: As 14 is a multiple of 7, we can denote that 224 is divisible by 7.

n divided by 5 yields a remainder equal to 3 is written as follows 
n = 5 k + 3 , where k is an integer.
add 2 to both sides of the above equation to obtain 
n + 2 = 5 k + 5 = 5(k + 1)
The above suggests that n + 2 divided by 5 yields a remainder equal to zero. 

Video solution:

5 and 12 it must a multiple of the lcm of 3, 5 and 12 which is 60. 
n = 60 k
n + 60 is also divisible by 60 since 
n + 60 = 60 k + 60 = 60(k + 1)

Video solution:

It is the lcm of 5, 7 and 20 which is 140. 

When n is divided by 8, the remainder is 3 may be written as 
n = 8 k + 3
multiply all terms by 6 
6 n = 6(8 k + 3) = 8(6k) + 18
Write 18 as 16 + 2 since 16 = 8 * 2. 
= 8(6k) + 16 + 2
Factor 8 out. 
= 8(6k + 2) + 2
The above indicates that if 6n is divided by 8, the remainder is 2.

We first expand (2n + 2)² 
(2n + 2)² = 4n ² + 8 n + 4
Factor 4 out. 
= 4(n ² + 2n + 1)
(2n + 2)² is divisible by 4 and the remainder is equal to 0.

 

Let xy be the whole number with x and y the two digits that make up the number. The number is divisible by 3 may be written as follows 
10 x + y = 3 k
The sum of x and y is equal to 9. 
x + y = 9
Solve the above equation for y 
y = 9 - x Substitute y = 9 - x in the equation 10 x + y = 3 k to obtain.
10 x + 9 - x = 3 k

Solve for x 

x = (k - 3) / 3

x is a positive integer smaller than 10 

Let k = 1, 2, 3, ... and select the first value that gives x as an integer. k = 6 gives x = 1 

Find y using the equation y = 9 - x = 8 

The number we are looking for is 18 .

 It is divisible by 3 and the sum of its digits is equal to 9 and it is the smallest and positive whole number with such properties.

Video solution:

we can also test for divisibilty by adding the digits and if the result is divisible by3 then the number is divisible by 3. 
3 + 3 + 9 = 15 , divisible by 3.
3 + 4 + 2 = 9 , divisible by 3.
5 + 5 + 2 = 12 , divisible by 3.
1 + 1 + 1 + 1 = 4 , not divisible by 3.
The number 1111 is not divisible by 3

Video solution:

To divisibility rule of 12 is that a number should be divisible by 4 and 3.
4 – The number is divisible by 4 as its last two digits end with 12 and 12 is the multiple of 4.
3 – To make a number divisible by 3 we need to add all its digits and the sum should be a multiple of 3.
2 + 9 + 6 + * + 1 + 2 = 20 + *
To make the sum a multiple of 3 we need to add 1 to it.So, 20  1 = 21 and hence 1 is the smallest number that should be placed instead of * to make it divisible by 12.
Therefore, the answer is Option 1: 1.

Step 1: 224, the numbers we get after separation are 22 and 4
Step 2: Multiply the unit digit with 2
2 x 4 = 8
Step 3: subtract the number from the rest of the number.
22 – 8 = 14
Step 4: As 14 is a multiple of 7, we can denote that 224 is divisible by 7.