(a) Consider the 7th row of Pascal's triangle. Observe that each interior number (that is, a number other than 1) is divisible by 7. For what values of n, for 1 ‰¤ n ‰¤ 12, are the interior numbers of the nth row divisible by n?
(b) Confirm that each of the values of n from part (a) is a prime number. Prove that, if p is a prime number, then each interior number of the pth row of Pascal's triangle is divisible by p.
(c) Show that, for any prime number p, the sum of the interior numbers of the pth row is 2p - 2.
(d) Calculate 2p - 2 for p = 7, and show that it is a multiple of 7.
(e) Use the results of parts (b) and (c) to show that, for any prime number p, 2p - 2 is a multiple of p.
In the following triangular table, known as Pascal's triangle, the entries in the nth row are the binomial coefficients

Observe that each number (other than the ones) is the sum of the two numbers directly above it. For example, in the 5th row, the number 5 is the sum of the numbers 1 and 4 from the 4th row, and the number 10 is the sum of the numbers 4 and 6 from the 4th row. This fact is known as Pascal's formula. Namely, the formula says that

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|6). (). (). . (C). . Oth row 1st row 1 2 1 13 3 1 2nd row 3rd row 1 4 6 4 1 4th row 15 10 10 5 1 5th row 16 15 20 15 6 1 6th row 1 7 21 35 35 21 7 1 7th row