Consecutively divides

It is easy to see, that none of the numbers M +1, M + 2, M + 3 and M + 4 can be prime (all 4 has to be composite).

The smallest positive integer M, which is followed by 4 consecutive composite integers is 23. However:

$M + 4 = 27 = 3^3 \text {which is not a factor in any of the previous integers in our sequence } \Rightarrow M ≠ 23$

Similarly, M≠24 since M + 3 = 27 won't necessarily divide N. $( \text {e.g. when } N = \frac {23!}{3^7} )$

The next smallest candidate for M is 31, as 32, 33, 34 and 35 are all composite.

$\text {In this case, } M + 1 = 31 + 1 = 32 = 2^5 \text { doesn't divide e.g. }$

$N = \frac {31!}{2^22} \text { but all the other integers from 1 to 31 divides it, }$

$\text { since it only has } 2^4 \text { in its prime factorisation instead of } 2^5.$

Now, M = 32 works, as 33, 34, 35 and 36 are all composites and it is easy to prove, that:

$\text {if } a, b, N \in \mathbb {Z} \text { ; a, b are coprimes ( gcd(a , b) = 1 ) }$

$\text { then } a | N , b | N \Rightarrow ab | N$

$M + 1 = 32 + 1 = 33 = 3 × 11$

$M + 2 = 32 + 2 = 34 = 2 × 17$

$M + 3 = 32 + 3 = 35 = 5 × 7$

$M + 4 = 32 + 4 = 36 = 4 × 9$

$\text {Hence, our answer should be: } \boxed {32}$

We are looking for the least value of M such that the set {M+1,M+2,M+3,M+4} contains no prime powers. If any of the four is a prime power, the least common multiple of {1,...,M} will not be divisible by it, in spite of being divisible by the first M numbers. As a starting point, we look for numbers M that are followed by four non-prime numbers: the first such number is 23, but while neither 25 nor 27 is prime, both are prime powers.
The next such number is 31: but in this case while neither 33 nor 35 is a prime power, 32 is. But if we take M = 32, then we see none of 33,34,35,36 is a prime power. Thus any number divisible by the least common multiple of 1 through 32 is also divisible by 33, 34, 35, and 36.

We see is N divisible by all primes <= M. So by saying M divides N we mean to say prime factors of M are a subset of N. So M+i divides N for all 1<i<4 implies there should not exist any prime between M and M+4. Also there should not exist any no. of the form a^b where a,b are natural nos otherwise when the no. is expressed in the form p^q where p is a prime and q is a natural no. q will exceed the highest power of the existing primes that divide N. So in that case the least possible value of M =32