Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:

Question

(a) are divisible by 5 and by 7.


(b) have distinct digits.


(c) are not divisible by either 5 or 7.

Answer 1

(a) div. by 5 and by 7 => div. by 35
We can find out that
35*286=10010 and 35*29=1015
Therefore the number divisible by 35, n35
= 286-29=257

(b) distinct digits
First digit has 9 choices (1-9)
second and subsequent digits 9,8,7 choices each
Numbers with distinct digits
= 9*9*8*7
= 4536

(c) not divisible by either 5 or 7
Divisible by 5, n5= (10000-1000)/5=1800
Divisisble by 7, n7 = (10003-1001)/7=1286
Divisible by 5 or 7 or both
=n5+n7-n35
=1800+1286-257
=2829

Numbers NOT divisible by either 5 or 7
=(10000-1000)-2829
=6171