1000th digit

N = 61218243036 N=61218243036\cdots Let N N be a very large number which is formed by the concatenation of multiples of 6. What will be the 1000 t h 1000th digit of this number when counted from left hand side?


The answer is 7.

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1 solution

Naren Bhandari
Oct 4, 2017

N = 612182430 \large N = 612182430\cdots

1.

Since N N is very large number that is formed by the concatenation of multiples of 6 6 . Counting all the multiples of 6 6 is tedious to find the 100 0 t h 1000^{th} digit in the large number N N .

In the number above 6 6 is the one digit number in N N . The two digit numbers that are the mutiples of 6 6 in between 10 10 and 100 100 initiate from 6 × 2 = 12 6\times2=12 and terminates at 6 × 16 = 96 6\times16 = 96 . Altogether there are 16 1 = 15 16-1 = 15 (excluding 6 6 being one digit number) two digits numbers and 15 × 2 = 30 + 1 = 31 15\times 2 = 30+1 = 31 digits in the concatenation from 1 1 to 100 100 that are multiple of 6 6 . Three digits numbers between 100 100 and 1000 1000 that are mutiples of 6 6 are 166 16 = 150 166-16 = 150 (excluding 1 one digit and 15 two digits number). Total digits are 150 × 3 + 31 = 481 150\times 3+31 = 481 in large number N N . To meet the 100 0 t h 1000^{th} digit in N N , remaining digits are 1000 481 = 519 1000-481 = 519 which are formed by the concatenation of four digits numbers. Here, 519 3 m o d ( 4 ) 519\equiv 3\mod(4) where 129 129 are pairs of four digits numbers. These 129 129 pairs of four digit numbers lies in between 167 167 and 300 300 . Since 519 519 (remaining pairs of 4 digits) is not divisible by 4 4 however, 520 = 1001 481 520=1001-481 or 516 = 997 481 516=997-481 is divisible by 4 4 which implies that 99 7 t h 997^{th} and 100 1 t h 1001^{th} digit will be 295 × 6 = 1770 295\times 6 = 1770 and 296 × 6 = 1776 296\times 6 = 1776 respectively. So 100 0 t h 1000^{th} digit left hand side of N = 612182430 17701776 \large N= 612182430\cdots17701776 will be 7 7 .

2.

Total one digit number that is multiple of 6 6 be w w . Then, w = 10 6 = 1 × 1 = 1 w = \left\lfloor \frac{10}{6}\right\rfloor = 1\times1 = 1 Total two digits numbers x = 100 6 1 = 16 1 = 15 2 x = 2 × 15 = 30 x =\left\lfloor\frac{100}{6}\right\rfloor-1 = 16-1 = 15 \Rightarrow 2x= 2\times15= 30

Total three digits numbers y = 1000 6 16 = 150 3 y = 450 y = \left\lfloor\frac{1000}{6}\right\rfloor -16 = 150 \Rightarrow 3y = 450

Concatenation of one, two and three digits that are multiple of 6 6 are 1 + 30 + 450 = 481 1+30+450=481 .

Since there are huge numbers of four digits between 1000 1000 and 10000 10000 digit numbers so such a mutiple of 6 6 should be obtaines very close to 100 0 t h 1000^{th} and to meet 100 0 t h 1000^{th} digit in the above concatenation required numbers of are 1000 481 = 519 1000-481= 519 that total four digits numbers ie 4 z = 519 z = 519 4 = 129 4z =519 \Rightarrow z = \left\lfloor\frac{519}{4}\right\rfloor = 129 pairs of four digits numbers that start from 167 = 167 × 6 = 1002 167 = 167\times 6 = 1002 to 296 × 6 = 1776 296\times 6 = 1776 that represents a pair of 4 4 from 99 7 t h 997^{th} to 100 1 t h 1001^{th} digits in concatenation of multiples of 6 6 .

Therefore, 100 0 t h 1000^{th} digits is 7 7 .

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