Question

A polygon is given. The vertices are labeled with natural numbers from 1 to 2000 in the increasing order. First, we cross out the
number 1 then numbers 16, 31, 46, ... until one number is crossed out two times. What is that number? How many numbers are
left uncrossed?
Г2
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2
A​

Answer 1

Observe the pattern repeatedly and arrive at the answer.

Step-by-step explanation:

Given: Vertices of a polygon are labeled from 1 to 2000. We cross out numbers in the order 1, 16, 31, 46....

Find: How many numbers are left uncrossed.

Solution:

The relation between 1, 16, 31, 46  is 1, 1 + 15, 1 + 15 + 15....  

So the number being cross are in the form 1+ 15k.

The first round of numbers being crossed will be 1, 16, 31, 46.... 1995.

We will continue to cross 11, 26, 41 to 1991.

Next we will cross out 6, 21, 36 to 1986.

You can observe the pattern and arrive at the answer.

Answer 2

Solution :-

Case 1) :- Numbers crossed are 1, 16, 31, 46 _________ tn

where,

→ tn = (15n + 1) ≤ 2000 .

→ a + (n - 1)d ≤ 2000

→ 1 + (n - 1)15 ≤ 2000

→ 15n - 14 ≤ 2000

→ 15n ≤ 2014

→ n ≤ 134

so,

→ tn = 15 * 133 + 1 = 1996 .

then,

→ Numbers left are :- 2,3,4,5,_____ 15,17, ____ 30,32, _____ 45,47 ________ 1995, 1997, 1998 , 1999 and 2000 .

case 2) :- Numbers crossed are 2, 17, 32, 47, ________ 1997

then,

→ Numbers left are :- 3, 4, 5, _____ 15,18,______ 30,33 _______ 1995, 1998 , 1999 and 2000 .

case 3) :- Numbers crossed are 3, 18, 33, 48, ________ 1998 .

then,

→ Numbers left are :- 4, 5, _____ 15,19______ 30,_______ 1995, 1999 and 2000 .

case 4) :- Numbers crossed are 4, 19, 34, 49, ________ 1999 .

then,

→ Numbers left are :- 5,6 _____ 15______ 30,_______ 1995 and 2000 .

case 5) :- Numbers crossed are :- 5, 20, 35, _______ 2000 .

then,

→ Numbers left are :- 6,7,8, _________ 1995 .

case 6) :- Numbers crossed are 6, 21 , 36, 51 _______ 1986

then,

→ Left numbers are :- 7,8 ________ 1995 .

Case 7) :-

• crossed :- 7, 22 , 37 _______ 1987 .
• Left :- 8, 9, 10, ________ 1995 .

similarly, now,

→ case 8) :- cross starts with 8 ends with 1988

→ case 9) :- cross starts with 9 ends with 1980 .

→ case 10) :- cross starts with 10 ends with 1990 .

→ case 11) :- starts with 11 and ends with 1991 .

→ case 12) :- starts with 12 and ends with 1992 .

→ case 13) :- cross starts with 13 ends with 1993 .

→ case 14) :- starts with 14 and ends with 1994 .

→ case 15) :- starts with 12 and ends with 1995 .

finally, in case (16) again now, we have to cross 16 again .

Hence, 16 is the number crossed 2 times .

therefore,

→ Left numbers to be uncrossed are = 15 case * each case 133 numbers were crossed = 2000 - 15 * 133 = 2000 - 1995 = 5 numbers .

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