To determine the greatest number of sewing kits Julia can make with 49 needles and 42 spools of thread, we need to find the greatest common divisor (GCD) of the two numbers, 49 and 42.
We can find the GCD by using the Euclidean algorithm, which involves repeated division.
First, we divide 49 by 42: \[ 49 = 42 \times 1 + 7 \] This tells us that 49 divided by 42 gives a remainder of 7.
Next, we apply the algorithm again, using 42 and the remainder 7: \[ 42 = 7 \times 6 + 0 \] Here, we see that the remainder is 0, indicating that we can stop. The last non-zero remainder was 7, which means: \[ \text{GCD}(49, 42) = 7 \]
Now, we know the greatest common divisor is 7, which means that Julia can divide the needles and spools of thread into 7 equal sets without any leftover materials.
Next, we'll find out how many needles and spools each kit will get. For the needles: \[ \frac{49 \text{ needles}}{7 \text{ kits}} = 7 \text{ needles per kit} \] And for the spools of thread: \[ \frac{42 \text{ spools of thread}}{7 \text{ kits}} = 6 \text{ spools per kit} \]
Thus, Julia can create a total of
\[ \boxed{7} \]
sewing kits. Each kit will contain 7 needles and 6 spools of thread.