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?
​

Answer 1

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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