Open Middle®
Directions: Using the digits 0-9 at most once, fill in the boxes to make the largest possible greatest common factor.
Hint
How can we tell if your three numbers have any relationship with each other?
How can we use our knowledge of multiples?
Answer
97 would be the largest possible GCF (using the numbers 97 and 485 or 97 and 582)
Source: Howie Hua
97 is the GCF
97
Can you provide steps?
yes…I will give you the answer.
97 is the answer
In case anyone else is wondering:
Start with some easy numbers.
gcf(360, 12) is obviously 12, because 12 is a factor of 360.
Same goes for things like gcf(980, 40): 40 is a factor of 980, so 40 is the gcf.
In general: If the smaller number is a factor of the greater number, the gcf is equal to the smaller number.
If we want the gcf to be as big as possible, we could try searching for a pair of numbers, where the second one is a two-digit number that is as big as possible and the first is a multiple of it.
The greatest two-digit-number is 99, which has the digit 9 repeated, so let’s try 98 and search for a three-digiit number that is a multiple of 98 and doesn’t repeat any digit. The three digit numbers that are multiples of 98 are 196, 294, 392, 490, 588, 686, 784, 882 and 980. All of them contain either an 8 or a 9, which would then be repeated (because they are already in 98).
Now let’s try 97 and its multiples:
194, 291, 388, 485, 582, 679, 776, 873, 970.
Two of them, 485 and 582, contain neither a 7 nor a 9, so they are valid solutions.
97