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

(a) are divisible by 5 and by 7.
(b) are divisible by 7.
(c) are even.
(d) are divisible by 5 but not by 7.

(a) There are 257 numbers which are divisible by 5 and by 7

(b) There are 1286 numbers which are divisible by 7

(c) There are 4500 even numbers

(d) There are 514 numbers which are divisible by 5 but not by 7

Step-by-step explanation:

(a) Since, LCM(5, 7) = 35,

Thus, the numbers divisible by 5 and by 7 between 1000 and 9999,

1015, 1050, 1085............, 9975

Which is an AP,

Having first term, a = 1015,

Common difference, d = 35,

Last term, l = 9975

If n be the number of terms,

(b) Similarly,  number divisible by 7,

1001, 1008, 1015............, 9996

Here, a = 1001, d = 7, l = 9996

(c) Even numbers between 1000 and 9999 inclusive

1000, 1002, 1004, 1006,........9998

Here, a = 1000, d = 2, l = 9998

(d) Number divisible by 5,

1000, 1005, 1010............, 9995

Here, a = 1000, d = 5, l = 9995

Hence, the number divisible by 5 but not by 7 = numbers divisible by 5 - number divisible by 7

= 1800 - 1286

= 514

step-by-step explanation: