HSE Solutions to Problem sets

 03 July 16:38   

    Factorise the afterward numbers. (note: I understand you didnt accept to, this is just for those who are curious)

    #$13\; =\; 13\; cdot\; 1$

    #$26\; =\; 13\; cdot\; 2$

    #$59\; =\; 59\; cdot\; 1$

    #$82\; =\; 41\; cdot\; 2$

    #$101\; =\; 101\; cdot\; 1$

    #$121\; =\; 11\; cdot\; 11$

    #$2187\; =\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3$

    Factorise using recursion.

    #$45\; =\; 3\; cdot\; 3\; cdot\; 5$

    #$4050\; =\; 2\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 5\; cdot\; 5$

    #$2187\; =\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3\; cdot\; 3$

    #Use the aloft aftereffect to bound plan out the numbers that still charge to be beyond out in the table below, alive 5 is the next prime:

    :$$

    egin

    X & 2_p & 3_p & X & 5 &X &7& X& X& X 11 & X & 13 & X& X& X&17 &X& 19& XX& X& 23 & X& 25 &X&X&X&29& X31 &X& X& X& 35 &X&37& X& X& X41 & X& 43 & X& X&X&47& X& 49& X\end

    

    :The next prime amount is 5. Because 5 is an bare prime number, and 5

    2. Acquisition all primes beneath 200.

    :The adjustment will not be categorical here, as it is too long. However, all primes beneath 200 are:

    2 3 5 7 11 13 17 19 23 29

    31 37 41 43 47 53 59 61 67 71

    73 79 83 89 97 101 103 107 109 113

    127 131 137 139 149 151 157 163 167 173

    179 181 191 193 197 199

    #$(-1)\; cdot\; (-5)mod\; =\; 5$

    #$3\; cdot\; 7\; mod\; =\; 21\; =\; 10$

    #$2^1\; =\; 2,\; 2^2\; =\; 4,\; 2^3\; =\; 8,\; 2^4\; =\; 16\; =\; 5$

    $2^5\; =\; 32\; =\; 10,\; 2^6\; =\; 64\; =\; 9,\; 2^7\; =\; 128\; =\; 7$

    $2^8\; =\; 256\; =\; 3,\; 2^9\; =\; 512\; =\; 6,\; 2^\; =\; 1024\; =\; 1$

    An easier list: 2, 4, 8, 5, 10, 9, 7, 3, 6, 1