The first six rows of Pascal's triangle are shown below, beginning with row 0.

How many of the first 100 rows contain at least one even entry and no odd entries other than ? (Rows 2 and 4 have this property, for example.)

I didn't show the first 6 rows below, but I figured you could find that out yourself.

ColdplayMX
Aug 21, 2018

#2**+1 **

How many of the first 100 rows contain at least one even entry and no odd entries other than 1 ??

I'm not sure, but looking at the patterns in the triangle for the first 20 rows shows that evey 2^{n} row [ for n ≥ 1 ] will only contain even entries [except for the beginning and ending "1" entries in these rows]

So....we should have 6 rows since 2^{6} = 64th row..... but 2^{7} = 128th row

CPhill
Aug 21, 2018

#3**+1 **

You are correct in saying 6 rows. Starting with row 0, the kth row has the numbers C(k, 0) C(k, 1) C(k, 2) ... C(k, k). For every number in the row to be even, k must only have even factors, so it must be a power of 2. Since the highest power of 2 under 100 is 2^6, 6 of the first 100 rows have only even numbers besides 1.

ColdplayMX
Aug 21, 2018