+0

# Another Pascal's triangle question.

+1
427
4
+196

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.

Aug 21, 2018

#1
+209
+1

Hmm... No clue :(

Aug 21, 2018
#2
+101161
+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 2n 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   26  = 64th row.....   but 27   = 128th row

Aug 21, 2018
edited by CPhill  Aug 21, 2018
#3
+196
+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.

Aug 21, 2018
#4
0

Sorry to be dim, but can you explain why, for every number in a row to be even, k must only have even factors?

Guest Aug 21, 2018