Posted by **carlton** on Sunday, April 8, 2012 at 2:04pm.

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

(a) are divisible by 5 and by 7.

(b) have distinct digits.

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

- discrete math -
**MathMate**, Sunday, April 8, 2012 at 3:47pm
(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

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