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        >>> bit_size(1023)
        10
        >>> bit_size(1024)
        11
        >>> bit_size(1025)
        11

    :param num:
        Integer value. If num is 0, returns 0. Only the absolute value of the
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        >>> byte_size(1 << 1023)
        128
        >>> byte_size((1 << 1024) - 1)
        128
        >>> byte_size(1 << 1024)
        129

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        >>> ceil_div(100, 10)
        10
        >>> ceil_div(1, 4)
        1

    :param num: Division's numerator, a number
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    r)�divmod)rr"�quanta�modrrrr Qsr )rrrc	Cs�d}d}d}d}|}|}|dkr\||}|||}}||||}}||||}}q|dkrl||7}|dkr|||7}|||fS)z;Returns a tuple (r, i, j) such that r = gcd(a, b) = ia + jbrrr)	rr�x�y�lxZlyZoa�ob�qrrr�extended_gcdis r+)r&�nrcCs(t||�\}}}|dkr$t|||��|S)z�Returns the inverse of x % n under multiplication, a.k.a x^-1 (mod n)

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    :param a_values: the a-values of the above equation
    :param modulo_values: the m-values of the above equation
    :returns: x such that x = a[i] (mod m[i]) for each i


    >>> crt([2, 3], [3, 5])
    8

    >>> crt([2, 3, 2], [3, 5, 7])
    23

    >>> crt([2, 3, 0], [7, 11, 15])
    135
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