Two factors, each with no digit greater than 5, have a product of 16,848. What is the largest possible 3-digit factor satisfying these conditions?
324
To find the largest possible 3-digit factor satisfying the given conditions, we need to determine the factors of 16,848 that meet the condition of having no digit greater than 5.
To start, let's factorize 16,848. We can do this by finding its prime factors:
16,848 ÷ 2 = 8,424
8,424 ÷ 2 = 4,212
4,212 ÷ 2 = 2,106
2,106 ÷ 2 = 1,053
Now, we have expressed 16,848 as the product of its prime factors: 2 × 2 × 2 × 3 × 13 × 17.
Next, we need to pair these prime factors to form two-digit factors. We start by pairing the smallest prime factor with the largest one, then the second smallest with the second largest, and so on.
Possible pairings:
(2 × 17) = 34
(2 × 13) = 26
(2 × 3) = 6
Now, we have three potential two-digit factors: 34, 26, and 6.
To determine the largest possible three-digit factor satisfying the conditions, we need to find the largest possible value for these two-digit factors while ensuring that no digit is greater than 5.
Starting with the largest two-digit factor, which is 34, we can see that it satisfies the condition since its digits (3 and 4) are both less than 5.
Therefore, the largest possible three-digit factor satisfying the given conditions is 34.
In summary, the largest possible three-digit factor satisfying the conditions is 34.