x /= commonDivisor;
    denominator /= commonDivisor;
    // We know gcd(x, denominator) = 1, and x * numerator / denominator is exact,
    // so denominator must be a divisor of numerator.
    return x * (numerator / denominator);
  }

  /*
   * binomial(biggestBinomials[k], k) fits in a long, but not binomial(biggestBinomials[k] + 1, k).
   */
  static final int[] biggestBinomials = {
    Integer.MAX_VALUE,

}

// ceilFrac takes a numerator and denominator representing a fraction
// and returns its ceiling. If denominator is 0, it returns 0 instead
// of crashing.
func ceilFrac(numerator, denominator int64) (ceil int64) {
	if denominator == 0 {
		// do nothing on invalid input
		return
	}
	// Make denominator positive
	if denominator < 0 {
		numerator = -numerator
		denominator = -denominator
	}

   * quantities.
   *
   * <p><b>Java 8+ users:</b> use {@link Long#divideUnsigned(long, long)} instead.
   *
   * @param dividend the dividend (numerator)
   * @param divisor the divisor (denominator)
   * @throws ArithmeticException if divisor is 0
   */
  public static long divide(long dividend, long divisor) {
    if (divisor < 0) { // i.e., divisor >= 2^63:
      if (compare(dividend, divisor) < 0) {

	cases := []struct {
		numerator, denominator, ceiling int64
	}{
		{0, 1, 0},
		{-1, 2, 0},
		{1, 2, 1},
		{1, 1, 1},
		{3, 2, 2},
		{54, 11, 5},
		{45, 11, 5},
		{-4, 3, -1},
		{4, -3, -1},
		{-4, -3, 2},
		{3, 0, 0},
	}
	for i, testCase := range cases {
		ceiling := ceilFrac(testCase.numerator, testCase.denominator)
		if ceiling != testCase.ceiling {

   * quantities.
   *
   * <p><b>Java 8+ users:</b> use {@link Integer#divideUnsigned(int, int)} instead.
   *
   * @param dividend the dividend (numerator)
   * @param divisor the divisor (denominator)
   * @throws ArithmeticException if divisor is 0
   */
  public static int divide(int dividend, int divisor) {
    return (int) (toLong(dividend) / toLong(divisor));
  }

  /**