Page 153 - Ganit Kaushal

Ex 20. Use the division algorithm to find the remainder when 78 is divided by

(a) 10 (b) 5 (c) 2
Sol. (a) By the division algorithm, division of 78 by 10 can be written as 78 = 10 × 7 + 8. \ Remainder = 8
(b) Similarly, 78 = 5 × 15 + 3. \ Remainder = 3
(c) 78 = 2 × 39 + 0. \ Remainder = 0.

Ex 21. Fill the grid with prime numbers only, so that the product of each row is the 75
number to the right of the row and the product of each column is the number 42
below the column. (NCERT) 102

Sol. Prime factorisation of 75 = 3 × 5 × 5 170 30 63
Prime factorisation of 42 = 2 × 3 × 7 5 5 3 75
Prime factorisation of 102 = 2 × 3 × 17 2 3 7 42
Using the prime factorisation of 75, 42, 102 and putting it wisely in the empty 17 2 3 102
grid, we get the adjoining grid. 170 30 63
Ex 22. Here (a) and (b) are two different factor trees for 60. Write the missing numbers.

(a) 60 (b) 60

6 10 30 ?

2 ? 5 ? 10 ?

? ?

Sol. (a) Here 60 = 6 × 10 and further 6 = 2 × 3 and 60 60
10 = 5 × 2. So, the missing numbers are 3 and
2. Therefore, the complete factor tree is given
in the adjoining figure (i). 6 10 30 2
(b) Here 60 = 30 × 2 and further 30 = 10 × 3 and
10 = 5 × 2. So, the missing numbers are 2, 3, 2 3 5 2 10 3
2 and 5. Therefore, the complete factor tree is
given in the adjoining figure (ii). 5 2
Fig. (i) Fig. (ii)
Ex 23. Which factors are not included in the prime factorisation of a composite number?
Sol. 1 and the number itself are not included in the prime factorisation of a composite number.

Ex 24. In 385*4, replace ‘*’ by a digit so that the number is divisible by 6.
Sol. Since the given number is an even number, for it to be divisible by 6, the number formed should
be divisible by 3 i.e., the sum of all digits should be divisible by 3. Sum of digits in 385*4
(leaving ‘*’) = 3 + 8 + 5 + 4 = 20.
If we add any of the digits 1, 4 or 7, then the sum of the digits is 21, 24, or 27, respectively, which are
divisible by 3. On replacing ‘*’ by 1, 4 or 7, the corresponding numbers formed are 38514, 38544, or
38574, and each of these numbers is divisible by 6.
Hence, ‘*’ can be replaced by the digits 1, 4 or 7.

Prime Time 151