I don't know how to solve these question;
1)I have to show that 2power16 divides 32!. find the highest power of 2 that divides 32!
2)aiso prove that there are exactly 4! no.s b/w 1000 &10000 that contain the digits 1,3,5 &7
One way to consider the two questions in the first exercise is to think about prime factorizations for the even factors in the numerator. In other words, just how many factors of 2 are up there?
There are obviously at least 16 factors of 2 in the numerator, to cancel with 16 factors of 2 in the denominator (2^16). So, yes, 2^16 will divide evenly.
\(\displaystyle 32!\text{ contains the factors: }\:1,2,3,4,\,\hdots,32\)
Every 2nd factor contains 22 and each contributes a factor of 2: 232=16 factors of 2.
But every 4th factor contains 22 and each contributes an additional 2: 432=8 more factors of 2.
\(\displaystyle \text{And every }8^{th}\text{ factor contains }2^3\text{ which contributes another 2: }\quad \frc{32}{8} \,=\,4\text{ more factors of 2.}\)
And every 16th factor contains 24 which contributes another 2: 1632−2 more factors of 2.
And every 32ndfactor contains 25 which contributes another 2: 3232=1 more factor of 2.
Hence, 32! contains: 16+8+4+2+1=31 factors of 2.
. . That is, 32! contains a factor of 231
Therefore, it is certainly divisible by 216.
2) Prove that there are exactly 4! nimbers between 1000 &10000 that contain the digits 1,3,5 and 7
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