Sagot :
• Problem:
What is the greatest common factor of
2 × 3 × 5 × 7 and 2 × 2 × 3 × 5 × 7?
• Solution:
The numbers are 210 and 420. Their greatest common factor is 210 while their least common multiple is 420.
[tex] \: \: \: \: \: \: \: \: \tt2 × 3 × 5 × 7 = 210 \\ \underline{ \tt 2 × 2 × 3 × 5 × 7 =420 } \\ \: \: \: \: \: \: \: \: \: \: \: \: \tt2 \times 2 \times 3 \times 5 \times 7 = 420 \: (LCM) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt 2 \times 3 \times 5 \times 7 = 210 \: (GCF)[/tex]
• Answer:
The greatest common factor of 2 × 3 × 5 × 7 and 2 × 2 × 3 × 5 × 7 is 210.
[tex] \large \color{indigo}{ \bold{PROBLEM:}}[/tex]
common factor of
What is the greatest common factor of
2 X3 X5 X7 and 2x2x3 X5X7
[tex] \huge \bold{ \blue{A}} \pink{n} \purple{s} \red{w} \green{e} \color{yellow}{r :} \\ \large \orange {\underline \bold{ \boxed{210}}}[/tex]
[tex] \large \bold{First \: we \: need \: to \: Multply}[/tex]
[tex] \large \bold{2 \times 3 \times 5 \times 7} = \large \underline{ \boxed{ \bold{210}}} \\ \underline{ \large \bold{2 \times 2 \times 3 \times 5 \times 7} =\large \underline{ \boxed{ \bold{420}}}}[/tex]
[tex] \bold{Now \: the \: GCF \: of \: 210 \: and \: 420 \: is\: 210}[/tex]
[tex] \bold{List\: of\: Factor :}[/tex]
• The factors of 210 are: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
• The factors of 420 are: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
The Greatest Common Factor is the largest of these numbers. For 210 and 420, it is: 210
[tex]\bold{Prime \: Factorizations : }[/tex]
• The prime factors of 210 are: 2, 3, 5, 7
• The prime factors of 420 are: 2, 2, 3, 5, 7
• List of all the common prime factors: 2, 3, 5, 7
• Find the product of all common prime factors: 2 * 3 * 5 * 7 = 210
• The Greatest Common Factor is the result of the previous step. For 210, and 420 is: 210
[tex] \bold{Euclidean\: Algorithm : }[/tex]
• Sort the numbers into ascending order:
210, 420
• Take the smallest number (210) as your divisor
• Work out the modulo operation of the remaining number(s) and the divisor:
420 mod 210 = 0
• Gather the divisor and all of the remainders and sort them in ascending order. Remove any duplicates and 0.
• As there is only one number left (the divisor), it's the Greatest Common Factor
• Therefore the Greatest Common Factor of 210 and 420 is: 210
[tex]\bold{Upside \:Down\: Division : }[/tex]
• Start by writing all of your numbers next to each other:
[tex] \large \bold{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 210 \: 420}[/tex]
• Find a prime number which can divide at least two of your numbers by (without remainder).
Write that prime number on the left hand side:
[tex] { \large {\: \: \: \: \: \: \: \: \: \: \: \: \: \colorbox{red}{ 2}} \large\bold{210 \: 420}}[/tex]
• Divide your original numbers by the prime and write the quotients under original numbers:
[tex] \large \bold{ \: \: \: \: \: \: \: \: \: \colorbox{red}2} \large \bold{ 210 \: 420} \\ \underline{\large \bold{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 105 \: 210}}[/tex]
• Repeat until the whole table is complete:
[tex] \underline{ \large \bold \colorbox{red}{2} \large \bold{210 \: 420}} \\ \underline{\large \bold \colorbox{red}{3} \large \bold{105 \: 210}} \\ \underline{ \large \bold \colorbox{red}{5} \large \bold{35 \: 70}} \\ \underline{\large\bold\colorbox{red}{7} \large \bold{7 \: 14}} \\ \large \bold{ \: \: \: \: \: \: \: 1 \: \: \: 2}[/tex]
• Find the GCF by multiplying all values in the orange column:
2 * 3 * 5 * 7 = 210
• The Greatest Common Factor of 210 and 420 is: 210
[tex] \large \bold{Binary\:(Stein's) \: Algorithm : }[/tex]
• Sort the numbers into ascending order:
210, 420
• Let your inital GCF equal 1.
• All of the numbers are even. Divide all of them by 2 and multiply your GCF by 2:
• 105, 210
• GCF = 1 * 2 = 2
• Divide all of the remaining even values by 2 (if there are any):
• 105, 105
• Sort the values into ascending order (if they aren't already). Remove any duplicates:
105
• There is only one number left, 105. Multiply it by your current GCF:
• GCF = 2 * 105 = 210
• Therefore the Greatest Common Factor of 210 and 420 is: 210
[tex]\purple{\begin{gathered} \gamma \\ \huge \boxed{ \ddot \smile}\end{gathered}} \: \pink{\begin{gathered} \gamma \\ \huge \boxed{ \ddot \smile}\end{gathered}} \: \red{\begin{gathered} \gamma \\ \huge \boxed{ \ddot \smile}\end{gathered}} \: \orange{\begin{gathered} \gamma \\ \huge \boxed{ \ddot \smile}\end{gathered}}[/tex]
[tex] \large \color{indigo} { \bold{Carry\:On \: Latex}}[/tex]