Consider a 5 digit number, for single digits non-negative integers .
If both values of are selected at random, what is the probability that the resultant number is divisible by ?
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For divisibility of 33 we should check the divisibity of 11 and 3. I think I need not to explain that for divisibility of 3 we should have ( a + b ) = 2 , 5 , 8 , 1 1 , 1 4 , 1 7 Also for divisibility of 11 we need ( 3 + 2 + b ) − ( a + 5 ) = 0 (Since, the given numbers are not so large that we can get a multiple of 11 as difference) So we get, a + 5 = 5 + b Or, a = b Or, a + b = 2 a = 2 b So a + b is even. Hence only 2 , 8 , 1 4 are required 3 sums and there are only three sets of a & b . The number of ways in which a & b can be chosen are 1 0 × 1 0 = 1 0 0 Therefore required probability = 1 0 0 3 = 0 . 0 3