Even a set can be honest, why not you?

I did this in a not so honest way...

Python 3

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def prod(numbers):
    result = 1
    for n in numbers:
        result *= n
    return result
def honest(numbers):
    return prod(numbers) % 840 != 0
count = 0
samples = 0
for i in range(1, 100 - 4):
    numbers = (i,i+1,i+2,i+3,i+4)
    if honest(numbers):
        count += 1
    samples += 1
from fractions import Fraction as frac
ratio = frac(count, samples)
answer = ratio.numerator+ratio.denominator
print (answer)

$840 = 2^3 × 3 × 5 × 7$ . Now, any 5-tuples of consecutive positive integers would have a multiple of 2, a multiple of 3, a multiple of 4 and a multiple of 5. So, the product of any 5 -tuples would be a factor of $2×3×4×5 = 2^3 × 3 × 5$ . So, the set of all honest positive integers is simply those 5-tuples which do not contain a multiple of 7 in it. Such 5-tuples 2 in every 7 of the consecutive sets that can be formed. So, the honest sets are those which start with 1, 2, 8, 9, 15, 16 and so on. There are $28$ such sets. And the total 5-tuples of consecutive no. each less than 100 which can be formed is 95. So, probability required = $\dfrac{28}{95}$ . So, $a + b = \boxed{123}$ .

Firstly,we note that the product of any 5 consecutive positive integers is always divisible by 8,3 and 5.Hence, for the product of any 5 consecutive integers to be not divisible by 840, none of the 5 numbers should be divisible by 7 (as 840=3×5×7×8).This is possible only if we have the first number, say x, (when the 5 consecutive integers are arranged in increasing order) satisfying the condition : x=1(mod 7) or x=2(mod 7).Keeping in mind that x is atmost 95 , the total solutions of x=1(mod 7) are {1,8,15,......,92} and that of x=2(mod 7) are {2,9,16,.......93}. Hence, total number of possible values of x are 28 values.Hence, favourable outcomes are 28.And, all possible values of x are {1,2,3,.....95} that are 95 total outcomes.Hence, the required probability is favourable number of outcomes divided by total number of outcomes = 28/95. So, our answer is 28+95=123.