Let

x = number of art kits

y = number of set of tools

We have to find x and y non-negative integers where 33x+29y = 2490

Using the Euclidean algorithm

33 = 1(29)+4 0 < 4 < 29

29 = 7(4) +1 0 < 1 < 4

4 = 4(1) + 0

Since 1 is the last nonzero remainder 33 and 29 are relative prime and:

1 = 29-7*4 =29-7(33-1*29) = 29-7*33 +7*29 = 8*29 -7*33

Then:

1 = 8*29 -7*33 multiply both sides by 2490

2490 = 33(-7 * 2490) + 29 (8*2490)

2490 = 33*(-17430) + 29*19920 + 33*29*k - 33*29*k = 33(29k-17430)+29(19920-33k) for any k integer

0 <= x = 29k - 17430 and 0 <= y = 19920 - 33k

Then:

17430 <= 29k and 33k <= 19920

17430/29 <= k <= 19920/33

601 < k <604

Then k = 602 or k = 603

1) For k = 602 then x = 29k-17430 = 28 and y = 19920 - 33k = 54

x = 28 and y = 54

2) For k = 603 then x = 29k-17430 = 57 and y = 19920 - 33k = 21

x =57 and y = 21

There are 2 options:

Option 1) She can buy 28 art kits and 54 tool sets.

Option 2) She can buy 57 art kits and 21 tool sets.