Page 71 - Ganit Kaushal

This result is true and can be verified by means of various examples. So, we generalise this result which is
known as the division algorithm.

If ‘a’ is a whole number and ‘b’ is another non-zero smaller whole number, then there exist unique whole
numbers q and r, such that:

a = (b × q) + r, where 0 ≤ r < b

Here, q is the quotient, r is the remainder, ‘a’ is the dividend and ‘b’ is the divisor.
If the remainder is 0, then ‘a’ is said to be divisible by ‘b’.

Ex 1. Divide (a) 4000 by 215, (b) 4500 by 25.
Also, write it in the form of the division algorithm.

Sol. (a) Using the long division method, we observe that: 1 8
215 4000
The dividend is 4000 (a), the divisor is 215 (b), the quotient is 18 (q), and the remainder – 215
1850
is 130 (r). – 1720
Therefore, according to the division algorithm, we can write: 4000 = 215 × 18 + 130 130

(b) Using the long division method, we observe that: 180
The dividend is 4500 (a), the divisor is 25 (b), the quotient is 180 (q), and the remainder 25 4500
– 25
is 0 (r). 200
– 200
Therefore, according to the division algorithm, we can write: 4500 = 25 × 180 + 0, 00
i.e., 4500 = 25 × 180.

Ex 2. Find the greatest 4-digit number that is exactly divisible by 136.
Sol. The greatest four-digit number = 9999. 73
136 9999
On dividing 9999 by 136, using the long division method, we get the remainder = 71. – 952
479
To get the greatest 4-digit number that is exactly divisible by 136, we subtract 71 from 9999. – 408
\ The required number = 9999 – 71 = 9928. 71
Ex 3. Find the least 4-digit number that is exactly divisible by 45.
Sol. The least four-digit number = 1000.

On dividing 1000 by 45, using the long division 22
method, we get the remainder = 10 45 1000 Note

To get the least 4-digit number that is exactly – 90 Here if we subtract 10 from 1000
100
divisible by 45, we add (45 – 10) = 35 to 1000. – 90 we get a three-digit number 990,
\ The required number = 1000 + 35 = 1035. 10 that is divisible by 45.

Ex 4. Find the least number that should be added to 1000 so that the sum is exactly divisible by 45.
Sol. Same as Example 3 above. The required number is 35.

Play with Digits

There are a total of 10 digits, namely, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these 10 digits we write counting numbers:
1, 2, 3, 4, and so on. These numbers are infinite (endless) in quantity.

Out of these counting numbers, there are only 9 one-digit numbers, from 1- 9.

Number Play 69