The number of elements in the set {n 
N: 10 
n 
100 and 3 n - 3 is a multiple of 7} is_______
Text Solution
Verified by Experts15
(15)
Given that n 
[10,100] and n 
N.
Let us take the values of n = 1,2, 3... and check whether 3 n - 3 is divisible by 7 or not.
3 1 - 3 = 0 is divisible by 7
3 2 - 3 = 3, 3 is not divisible by 7
3 6 = 7k+l is not divisible by 7
3 7 = 7 
+ 3,
3 7 - 3 = 7 
is divisible by 7
Similarly, 
3 13 = (7k + l) (7 
+ 3)
= 7 
+ 3
3 13 - 3 = 7 
is divisible by 7
If we observe the pattern for n = 1,7,13, 3 n -3 is divisible by 7.
That means the series forms an AP with d = 6.
Also, n 
[10,100]
The required progression is 13,19, ......a n
a n = 13+(n- 1)6
But 13 +(n - 1)6 < 100
n< 15.5
n = 15 
N
Therefore, the required answer is 15.
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