(g) Prove that 11 divides a if and only if 11 divides the alternating sum ao - a1 + a2 -.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Just need g and h
![6. Let a be a number written in base 10 as
x = ao + a110 +..a, 10"
where 0< a; < 10.
(a) Prove that 2 divides a if and only if 2 divides ao.
(b) Prove that 4 divides a if and only if 4 divides ao +2a1.
(c) Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2.
(d) Prove that 5 divides a if and only if 5 divides ao.
(e) Prove that 9 divides a if and only if 9 divides the sum ao + ...a, of its digits.
(f) Prove that 3 divides a if and only if 3 divides the sum ao + ...an of its digits.
(g) Prove that 11 divides a if and only if 11 divides the alternating sum ao – a1 + az
....
(h) Come up with a rule for dividing by 7 and prove that it works.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2Fda1df587-56a3-4db6-b972-e96b1f4ed4b5%2F2a0na5w_processed.jpeg&w=3840&q=75)
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