we can use the arithmetic sequence :

an = a + (n - 1) . d

an = the nᵗʰ term in the sequence

a = the first term in the sequence

d = the difference between terms

answer :

first, we have to know, which number between 1000 and 2005 is divisible by 13. find the first term in the sequence.

1000 ÷ 13 = have a remainder

1001 ÷ 13 = have a remainder of 0

1001 is the first term of the sequence

and, we have to know the last term of the sequence.

2005 ÷ 13 = have a remainder

2004 ÷ 13 = have a remainder

2003 ÷ 13 = have a remainder

2002 ÷ 13 = have a remainder of 0

2002 is the last term of the sequence

now, find how many terms are there in the sequence. use the arithmetic sequence formula.

an = a + (n - 1) . d

2002 = 1001 + (n - 1) . 13

2002 - 1001 = 13n - 13

1001 = 13n - 13

13n = 1001 + 13

13n = 1014

n = 78

thus, there are 78 integers between 1000 and 2005 that is divisible by 13