library SafeMath {

  /**
   * @dev Multiplies two numbers, throws on overflow.
   * @param _a Factor number.
   * @param _b Factor number.
   */
  function mul(
    uint256 _a,
    uint256 _b
  )
    internal
    pure
    returns (uint256)
  {
    if (_a == 0) {
      return 0;
    }
    uint256 c = _a * _b;
    assert(c / _a == _b);
    return c;
  }

  /**
   * @dev Integer division of two numbers, truncating the quotient.
   * @param _a Dividend number.
   * @param _b Divisor number.
   */
  function div(
    uint256 _a,
    uint256 _b
  )
    internal
    pure
    returns (uint256)
  {
    uint256 c = _a / _b;
    // assert(b > 0); // Solidity automatically throws when dividing by 0
    // assert(a == b * c + a % b); // There is no case in which this doesn't hold
    return c;
  }

  /**
   * @dev Substracts two numbers, throws on overflow (i.e. if subtrahend is greater than minuend).
   * @param _a Minuend number.
   * @param _b Subtrahend number.
   */
  function sub(
    uint256 _a,
    uint256 _b
  )
    internal
    pure
    returns (uint256)
  {
    assert(_b <= _a);
    return _a - _b;
  }

  /**
   * @dev Adds two numbers, throws on overflow.
   * @param _a Number.
   * @param _b Number.
   */
  function add(
    uint256 _a,
    uint256 _b
  )
    internal
    pure
    returns (uint256)
  {
    uint256 c = _a + _b;
    assert(c >= _a);
    return c;
  }

}
/**
 * @dev Math operations with safety checks that throw on error. This contract is based
 * on the source code at https://goo.gl/iyQsmU.
 */
function mul(
  uint256 _a,
  uint256 _b
)
  internal
  pure
  returns (uint256)
{
  if (_a == 0) {
    return 0;
  }
  uint256 c = _a * _b;
  assert(c / _a == _b);
  return c;
}

/**
 * @dev Multiplies two numbers, throws on overflow.
 * @param _a Factor number.
 * @param _b Factor number.
 */