3. (10 points) $[a]$ represents the greatest integer not greater than $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{k}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, then the largest positive integer $k$ not exceeding 48 is $\qquad$
3. (10 points) $[a]$ represents the greatest integer not greater than $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ ( $\left.\left[\frac{k}{7}\right]+1\right)$ when divided by 13 leaves a remainder of 7, then the largest positive integer $k$ not exceeding 48 is 45.
【Analysis】 The remainder of the product of $a$ and $b$ when divided by $c$ is equal to the product of the remainders of $a$ and $b$ when divided by $c$ (or the remainder of this product when divided by $c$). For example, the remainders of 23 and 16 when divided by 5 are 3 and 1, respectively, so the remainder of $(23 \times 16)$ when divided by 5 is $3 \times 1=3$. Note: When the product of the remainders is greater than the divisor, the required remainder is the remainder of the product of the remainders when divided by $c$. For example, the remainders of 23 and 19 when divided by 5 are 3 and 4, respectively, so the remainder of $(23 \times 19)$ when divided by 5 is the remainder of $(3 \times 4)$ when divided by 5. This problem can be analyzed based on this property.
【Solution】Solution:
The product of the first 6 numbers is $1 \times 1 \times 1 \times 1 \times 1 \times 1=1$;
The product of the 7th to the 13th numbers is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=128 ; 128 \div 13=9 \cdots 11$;
The product of the 14th to the 20th numbers is $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3=2187,2187 \div 13=168 \cdots 3 ; 11 \times 3$ $=33,33 \div 13=2 \cdots 7$;
The product of the 21st to the 27th numbers is $4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4=16384,16384 \div 13=1260 \cdots 4 ; 4$ $\times 7=28,28 \div 13=2 \cdots 2 ;$
The product of the 28th to the 34th numbers is $5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5=78125,78125 \div 13=6009 \cdots 8 ; 2$ $\times 8=16,16 \div 13=1 \cdots 3 ;$
The product of the 35th to the 41st numbers is $6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6=279936,279936 \div 13=21533 \cdots 7$; $3 \times 7=21,21 \div 13=1 \cdots 8 ;$
Since $8 \times 7 \times 7 \times 7 \times 7=19208,19208 \div 13=1477 \cdots 7$, $k$ is 45.