The least number which is a multiple of 11 and when divided by 3, 5 an...
The L.C.M. of 3, 5 and 9 is 45.
Hence, the number is of the form (45a + 2).
The least value of ‘a’ for which (45a + 2) is multiple of 11 is 9.
Therefore, the number is 407.
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The least number which is a multiple of 11 and when divided by 3, 5 an...
Problem Statement:
Find the least number which is a multiple of 11 and when divided by 3, 5 and 9 leaves 2 as a remainder.
Solution:
Let the required number be x.
As the number is a multiple of 11, we can write:
x = 11k, where k is an integer.
Now, let's apply the given conditions:
x leaves a remainder of 2 when divided by 3.
x leaves a remainder of 2 when divided by 5.
x leaves a remainder of 2 when divided by 9.
We can write these conditions as:
x = 3a + 2, where a is an integer.
x = 5b + 2, where b is an integer.
x = 9c + 2, where c is an integer.
Substituting the value of x from the first equation in the other two equations, we get:
11k = 3a + 2, and 11k = 5b + 2.
Solving these equations, we get:
a = (11k - 2)/3, and b = (11k - 2)/5.
As 'a' and 'b' are integers, (11k - 2) should be divisible by both 3 and 5.
The smallest such number is 15, so we can write:
11k - 2 = 15m, where m is an integer.
Solving for 'k', we get:
k = (15m + 2)/11.
As 'k' is an integer, the smallest value of 'm' for which this is true is 17.
Substituting this value in the above equation, we get:
k = 29.
Therefore, the required number is:
x = 11k = 11*29 = 319.
Hence, the correct option is (C) 407.