1. Find dimensions of the smaller plots:
   • Since each smaller plot has an area of $120 square meters120\; \backslash text\{\; square\; meters\}$120 square meters, we need to find pairs of integers $(a,b)(a,\; b)$(a,b) such that $a\times b=120a\; \backslash times\; b\; =\; 120$a×b=120.

   The factors of 120 and their corresponding pairs are:

   1×1202×603×404×305×246×208×1510×1212×1015×820×624×530×440×360×2120×1​=120=120=120=120=120=120=120=120=120=120=120=120=120=120=120=120​

2. Check compatibility with the field dimensions:
   • We need to check if the field of 60 meters by 40 meters can be divided into smaller plots of dimensions $a\times ba\; \backslash times\; b$a×b.

   To fit the dimensions into the field exactly, the length and width of the field (60 meters and 40 meters) must be divisible by the dimensions of the plot.

3. Filter valid plot dimensions:
   • For each pair $(a,b)(a,\; b)$(a,b), check if both 60 is divisible by $aa$a and 40 is divisible by $bb$b, or vice versa.

   Valid pairs:

   (a,b)(a,b)(a,b)(a,b)(a,b)(a,b)(a,b)(a,b)​=(10,12)=(12,10)=(20,6)=(6,20)=(15,8)=(8,15)=(30,4)=(4,30)​(since 60÷10=6 and 40÷12=3.333, not valid)(since 60÷12=5 and 40÷10=4, valid)(since 60÷20=3 and 40÷6=6.666, not valid)(since 60÷6=10 and 40÷20=2, valid)(since 60÷15=4 and 40÷8=5, valid)(since 60÷8=7.5 and 40÷15=2.666, not valid)(since 60÷30=2 and 40÷4=10, valid)(since 60÷4=15 and 40÷30=1.333, not valid)​So, valid pairs that fit exactly into the 60 by 40 field are:

   • $(12,10)(12,\; 10)$(12,10)
   • $(6,20)(6,\; 20)$(6,20)
   • $(15,8)(15,\; 8)$(15,8)
   • $(30,4)(30,\; 4)$(30,4)
4. Counting the number of ways:
   • There are four distinct pairs of dimensions that can be used to divide the field into smaller plots with integer dimensions.

Thus, there are $\boxed{{\displaystyle 4}}\backslash boxed\{4\}$4​ different ways the farmer can divide the field into smaller plots of 120 square meters each with integer dimensions.