Please show me how you did it. Thanks for your help.
4 Answers

If you think of seven to its early powers you will see the mod: (remainder when divided by 6)
7/6 has R:1
49/6 has R:1
343/6 has R:1 and
2401/6 has R:1
7 to any power in mod 6 has a remainder of 1
Since 7 is 6 with a remainder of 1 we can treat 7^1000 mod 6 as 1^1000 mod 6 which is 1 divided by 6 which of course has remainder 1.
Similarly for 6^1001 mod 7, if we divide the early powers of 6 by 7 we see the pattern in the remainders.
6/7 has remainder 6
36/7 has remainder 1
216/7 has remainder 6
1296/7 has remainder 1
We see that 6 to an odd power divided by 7 has remainder 6
and 6 to an even power divided by 7 has remainder 1
so 6^1001 mod 7 being an odd power will have a remainder of 6.
We could say that 6 in mod 7 is 1
so 1^1001 mod7 = 1 mod 7….and since 71=6 the answer is 6.
In short
7^1000 mod 6 = 1^1000 mod 6 = 1 mod6 = 1
6^1001 mod 7 = (1)^1001 mod 7 = 1 mod 7 = 71 = 6

The remainder of the products equals the remainder of the product of the remainders.
Examples
6 mod 5 = 1
8 mod 5 = 3
(68) mod 5 = (13) mod 5 = 3
11 mod 7 = 4
1111 mod 7 = (44) mod 7 = 2
Now lets try the first one:
7 mod 6 = 1
7^1000 mod 6 = 1^1000 mod 6 = 1
The second one is more difficult:
6 mod 7 = 6
The above law results in the same expression.
6^1001 = 6^1001 mod 6
Let’s write 6^1001 a different way:
6^1001 = (6^1000)6 = (36^500)6 mod 6
36 mod 7 = 1
36^500 mod 7 = 1^500 mod 7 = 1
(36^500)6 = (16) mod 7 = 6

7^1000 mod 6 = 1^1000 mod 6 = 1.
6^1001 mod 7 =(1)^1001 mod 7 = 1 = 6.

7^1000 mod 6 is 7 mod 6 done a 1000 times or 1 * 1000 =1
and so forth.