Division of Large Numbers
Division of two numbers can be treated as repeated subtraction. It is subtracting the smaller number from the greater number repeatedly till you reach zero or get a remainder.
Examples-
(a) 20 ÷ 4 = 5
20 – 4 – 4 – 4 – 4 – 4 = 0 i.e. by 5 times subtracting 4 from 20.
(b) 18 ÷ 6 = 3
18 – 6 – 6 – 6 = 0 i.e. by 3 times subtracting 6 from 18.
Now, suppose John has 2040 chocolates and he wants to divide them among 170 students. He wants to find out how many chocolates each would get. It is not possible for him to repeatedly subtract 170 from 2040 and find the quotient as the numbers are large and it would be a time taking process.
So, how can he divide to find the number of chocolates for each student?
Jared uses Long Division Method to find the solution. Let us have a look-
In this method,
· We start dividing left most digit of the dividend by the divisor and remainder is calculated.
· This remainder is carried forward and is followed by next digit of the number brought down from the dividend.
· This process continues until remainder is 0 or not divisible further.
So, we can say that each student would get 12 chocolates.
We use long division method when the divisor is of two or more digits to make the calculations easier and faster.
Let us look at practical problem on division of large numbers-
Example-
The cost of 125 Play Stations is $31,625. Find the cost of 1 Play Station.
Solution-
Cost of 125 Play Stations = $31,625
Cost of 1 Play Station = 31625 ÷ 125
So, the cost of 1 Play Station is $253.
Properties of Division
Properties help us to make our calculations easier, faster and reliable when used at appropriate places. Let us have a look-
· Division by 0: Division by 0 is not possible.
o Example: 51,841 ÷ 0 = Not Possible
· Division by 1: If a number is divided by 1, the result is the number itself.
o Example: 859,283 ÷ 1 = 859,283
· Division by number itself: If a number is divided by itself, the result is always 1.
o Example: 455,993 ÷ 455,993 = 1
· Dividing 0: If 0 is divided by any number (except 0), the result is always 0.
o Example: 0 ÷ 493,661 = 0
· Division Algorithm-
o Dividend = Divisor × Quotient + Remainder
o This equation is used to check division.
o Example- 725 ÷ 16
Divisor = 16
Quotient = 45
Remainder = 5
So,
Divisor × Quotient + Remainder
= 16 × 45 + 5 = 720 + 5 = 725 = Dividend
Division of a number by 10, 100, 1000, 10000
1. When a number is divided by 10, we simply remove the last digit to get the quotient. The last digit of the number forms the remainder.
2. When a number is divided by 100, we simply remove the last two digits to get the quotient. The last two digits of the number forms the remainder.
3. When a number is divided by 1000, we simply remove the last three digits to get the quotient. The last three digits of the number forms the remainder.
4. When a number is divided by 10000, we simply remove the last four digits to get the quotient. The last four digits of the number forms the remainder.
Examples-
· 7857 ÷ 100
· 28345 ÷ 10000
How do we know if a number is odd or even? If the number ends in 0, 2, 4, 6 or 8 we say it is even, else it is odd.
How do we know if one number is completely divisible by another without actually dividing them? We can find that out using Divisibility Tests.
Divisibility tests are used to find whether one number is divisible by another without carrying any actual division. Let us look at the basic tests of divisibility
Basic Tests of Divisibility
· Test of Divisibility by 2- For a number to be divisible by 2, the unit digit of the number should be 0, 2, 4, 6 or 8 i.e. the number should be even.
Example-
· 658- is divisible by 2 because its last digit is even.
Hence, proved.
659- is not divisible by 2 because its last digit is odd.
Hence, proved.
· Test of Divisibility by 3- A number is divisible by 3, if the sum of its digit is divisible by 3.
Example-
· 504- Sum of Digits = 5 + 0 + 4 = 9
And, 9 ÷ 3 = 3
Since, 9 is divisible by 3, so, 504 is also divisible.
Hence, proved.
· 505- Sum of Digits = 5 + 0 + 5 = 10
Now, 10 ÷ 3 = 3 & Remainder = 1
Since, 10 is not divisible by 3, so, 505 is also not divisible.
Hence, proved.
· Test of Divisibility by 4- A number is divisible by 4, if the last two digits of the number is divisible by 4.
Example-
· 92616- Last two digits = 16
And, 16 ÷ 4 = 4
Since, 16 is divisible by 4, so, 92616 is also divisible.
Hence, proved.
· 16733- Last two digits = 33
Now, 33 ÷ 4 = 8 & Remainder = 1
Since, 33 is not divisible by 4, so, 16733 is also not divisible.
Hence, proved.
· Test of Divisibility by 5- A number is divisible by 5, if the last digit of the number is either 0 or 5.
Example-
· 85015- Last digit = 5
Since, the last digit of the number is 5, so, 92615 is divisible by 5.
Hence, proved.
· 14819- Last digit = 9
Since, the last digit of the number is not 0 or 5, so, 14819 is not divisible by 5.
Hence, proved.
· Test of Divisibility by 6- A number is divisible by 6, if the number is divisible by both 2 & 3.
Example-
· 78696- Even Number.
Sum of Digits = 7 + 8 + 6 + 9 + 6 = 36
And, 36 ÷ 3 = 12
Since, the number is even and sum of its digits is divisible by 3, so the number is divisible by both 2 & 3. Thus, 78696 is divisible by 6.
Hence, proved.
· 653408- Even Number
Sum of Digits = 6 + 5 + 3 + 4 + 0 + 8 = 26
Now, 26 ÷ 3 = 8 & Remainder = 2
Since, the number is even but sum of its digits is not divisible by 3, so the number is divisible by 2 but not 3. Thus, 653408 is not divisible by 6.
Hence, proved.
· Test of Divisibility by 7- For a number is divisible by 7, first we take the last digit of the number and multiply it by 2 and then we subtract this number from the remaining digits of the number. If the subtracted number is 0 or is divisible by 7, then the actual number is also divisible by 7.
Example-
· 784- Last Digit × 2 = 4 × 2 = 8
Remaining Digits – Last Digit = 78 – 8 = 70
And, 70 ÷ 7 =10
So, 784 is also divisible by 7.
Hence, proved.
· 189- Last Digit × 2 = 9 × 2 = 18
Remaining Digits – Last Digit = 18 – 18 = 0
So, 189 is divisible by 7.
Hence, proved.
· Test of Divisibility by 8- A number is divisible by 8, if the last three digits of the number is divisible by 8.
Example-
· 101008- Last three digits = 008
And, 008 ÷ 8 = 1
Since, 008 is divisible by 8, so, 101008 is also divisible.
Hence, proved.
· 291801- Last three digits = 801
Now, 801 ÷ 8 = 100 & Remainder = 1
Since, 801 is not divisible by 8, so, 291801 is also not divisible.
Hence, proved.
· Test of Divisibility by 9- A number is divisible by 9, if the sum of its digit is divisible by 9.
Example-
· 27504- Sum of Digits = 2 + 7 + 5 + 0 + 4 = 18
And, 18 ÷ 9 = 2
Since, 18 is divisible by 9, so, 27504 is also divisible.
Hence, proved.
· 51805- Sum of Digits = 5 + 1 + 8 + 0 + 5 = 19
Now, 19 ÷ 9 = 2 & Remainder = 1
Since, 19 is not divisible by 9, so, 51805 is also not divisible.
Hence, proved.
· Test of Divisibility by 10- A number is divisible by 10, if the last digit of the number is 0.
Example-
· 54080- Since the last digit of the number is 0, so, 54080 is divisible by 10
Hence, proved.
· 63428- Since the last digit of the number is not 0, so, 63428 is not divisible by 10
Hence, proved.
· Test of Divisibility by 11- For a number is divisible by 11, first we sum up the digits of the number at odd places and at even places respectively, then we find the difference of both. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
Example-
· 67287- Sum of Digits at Odd Places = 6 + 2 + 7 = 15
Sum of Digits at Even Places = 7 + 8 = 15
Now, Difference = 15 – 15 = 0
So, 67287 is divisible by 11.
Hence, proved.
· 93415- Sum of Digits at Odd Places = 9 + 4 + 5 = 18
Sum of Digits at Even Places = 3 + 1 = 4
Now, Difference = 18 – 4 = 14
And, 14 ÷ 11 = 1 & Remainder = 3
Since, 14 is not divisible by 11, so, 93415 is also not divisible by 11.
Hence, proved.
· Test of Divisibility by 12- A number is divisible by 12, if the number is divisible by both 3 & 4.
Example-
· 21696-
Sum of Digits = 2 + 1 + 6 + 9 + 6 = 24
And, 24 ÷ 3 = 8
Also, Last two digits ÷ 4 = 96 ÷ 4 = 24
Since, the number is divisible by both 3 & 4, so the number is also divisible by 12.
Hence, proved.
· 41742-
Sum of Digits = 4 + 1 + 7 + 4 + 2 = 18
And, 18 ÷ 3 = 6
Also, Last two digits ÷ 4 = 42 ÷ 4 = 10 & Remainder = 2
Since, the number is divisible by both 3 but not 4, so the number is not divisible by 12.
Hence, proved.