Explain how to find the greatest number of identical arrangements that can be made with 54 roses and 42 tulips with no flowers left over.

Answer:   GCF(54, 42) = 6 arrangementsStep-by-step explanation:The number of such arrangements will be the greatest common factor (GCF) of the two numbers. You recall from your times tables that ...   54 = 6×9   42 = 6×7The factors 9 and 7 have no factors in common, so 6 is the greatest common factor of 54 and 42.   6 identical arrangements of 9 roses and 7 tulips can be made._____Additional commentWhen the numbers are small and found in your memorized times tables, finding the GCF is done by consulting your memory. Otherwise, it can be done by factoring, or by using Euclid's algorithm.When the numbers are factored to primes, the GCF is the product of the prime factors the numbers have in common.   54 = 2×3×3×3   42 = 2×3×7These factorizations have 2×3 = 6 in common.__Euclid's algorithm has you find the remainder from the division of the larger number by the smaller. When that remainder is not zero, it replaces the larger number, and the process repeats. When that remainder is zero, the divisor is the GCF.   54/42 = 1 r 12   42/12 = 3 r 6   12/6 = 2 r 0 . . . . GCF = 6This algorithm works on numbers of any size (including fractions and mixed numbers), so is especially useful when the factorization is not obvious.