Answer the following [8 Marks]
a) Find the least number which is divisible by both 7 and 12.
b) If a number 34A8 is divisible by 11, find the value of A.
c) Check whether 9048 is divisible by both 6 and 8.
a) Find the least number which is divisible by both 7 and 12.
b) If a number 34A8 is divisible by 11, find the value of A.
c) Check whether 9048 is divisible by both 6 and 8.
a) The lowest number which will be divisible by both of these numbers is the LCM of these numbers.
Before finding the LCM of the numbers, firstly let us check whether the numbers are co-prime or not. To find whether the numbers are co-prime, we will find out their factors.
The factors of 7: 1, 7
The factors of 12: 1, 2, 3, 4, 6, 12
Since the only common factor is 1, the given two numbers are co-prime. The LCM of co-prime numbers is their product.
∴ The required number is 7×12 = 84. (2 marks)
b) To check the divisibility of a number by 11, the rule is to find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
Consider 34A8,
The digits at the even places are 4 and 8. [0.5 marks]
The digits at the odd places are 3 and A. [0.5 marks]
Sum of digits at even places = 4 + 8 = 12 [0.5 marks]
Sum of digits at odd places = A + 3 [0.5 marks]
Let us find the difference now.
A + 3 - 12 = 0
A - 9 = 0 or A - 9 = 11
A = 9 or A = 20, A = 20 is not possible.
So, the number is 3498. (1 mark)
c) If a number is divisible by both 2 and 3, then it will be divisible by 6 also.
Divisibility rule of 2:
9048 is divisible by 2 as it end with one of these (0, 2, 4, 6, 8) numbers or simply as it is an even number. [0.5 Mark]
9048 is divisible by 2 as it ends with 8.
Divisibility by 3:
If the sum of the digits is a multiple of 3, then the number is divisible by 3.
⇒9+0+4+8=21
Here 21 is divisible by 3 hence 9048 is divisible by 3. [1 Mark]
So, these two numbers are divisible by both 2 and 3 and hence it is divisible by 6. [0.5 Mark]
Divisibility rule of 8:
A number will be divisible by 8 only if the last three digits are divisible by 8.
The last 3 digits of 9048, i.e. 048 is divisible by 8.
48 ÷ 8 = 6
[1 Mark]
Therefore, 9048 is divisible by both 6 and 8.
Before finding the LCM of the numbers, firstly let us check whether the numbers are co-prime or not. To find whether the numbers are co-prime, we will find out their factors.
The factors of 7: 1, 7
The factors of 12: 1, 2, 3, 4, 6, 12
Since the only common factor is 1, the given two numbers are co-prime. The LCM of co-prime numbers is their product.
∴ The required number is 7×12 = 84. (2 marks)
b) To check the divisibility of a number by 11, the rule is to find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
Consider 34A8,
The digits at the even places are 4 and 8. [0.5 marks]
The digits at the odd places are 3 and A. [0.5 marks]
Sum of digits at even places = 4 + 8 = 12 [0.5 marks]
Sum of digits at odd places = A + 3 [0.5 marks]
Let us find the difference now.
A + 3 - 12 = 0
A - 9 = 0 or A - 9 = 11
A = 9 or A = 20, A = 20 is not possible.
So, the number is 3498. (1 mark)
c) If a number is divisible by both 2 and 3, then it will be divisible by 6 also.
Divisibility rule of 2:
9048 is divisible by 2 as it end with one of these (0, 2, 4, 6, 8) numbers or simply as it is an even number. [0.5 Mark]
9048 is divisible by 2 as it ends with 8.
Divisibility by 3:
If the sum of the digits is a multiple of 3, then the number is divisible by 3.
⇒9+0+4+8=21
Here 21 is divisible by 3 hence 9048 is divisible by 3. [1 Mark]
So, these two numbers are divisible by both 2 and 3 and hence it is divisible by 6. [0.5 Mark]
Divisibility rule of 8:
A number will be divisible by 8 only if the last three digits are divisible by 8.
The last 3 digits of 9048, i.e. 048 is divisible by 8.
48 ÷ 8 = 6
[1 Mark]
Therefore, 9048 is divisible by both 6 and 8.
