Question

How many 5-digit numbers are there that are divisible by either 60 or 70 but are not divisible by 120?

Answer 1

1820

Explanation:

As long as the 5-digit numbers are divisible by 60 they are also divisible by 120. See solution for the number of 5 digits number divisible by 60 and the solution of 5 digits divisible by 120.

The number of 5 digit number divisible by 60 and 70 is 215

The total 5-digit numbers that are divisible by either 60 or 70 = 1499+1286-215 = 2570

The total 5-digit numbers that are divisible by either 60 or 70 but are not divisible by 120:

(5 digit number divisible by 60 or 70) -

(5 digit number divisible by 120)

2570 - 750 = 1820

Find attached for solution

Answer 2

The number of 5-digit numbers divisible by either 60 or 70 but are not divisible by 120 is 2035, 5-digit numbers

Explanation:

Here, to find the number of 5-digit numbers divisible by 60 and 70 but not divisible by 120 we have

The 5-digit numbers are between 10000 and 99999

Therefore, we have;

`(99999)/(60) = 1666(13)/(20)` and
`(10000)/(60) = 166(2)/(3)`

Thus we have multiples of 60 from 167×60 to 1666×60 which are 5-digit numbers

Hence we have, 1666 - 167 = 1499 numbers which are 5-digits and divisible by 60

Similarly for 70, we have;

`(99999)/(70) = 1428(39)/(70)` and
`(10000)/(70) = 142(6)/(7)`

Therefore the total number of 5-digit numbers divisible by 70 is 1428 - 143 = 1285 5-digit numbers

Hence, the number of 5-digit numbers divisible by both 60 or 70 = 1499 + 1285 = 2784 5 digit numbers

The number of 5-digit numbers divisible by 120 is found similarly as follows;

`(99999)/(120) = 833(13)/(40)` and
`(10000)/(120) = 83(1)/(3)` which is 833 - 84 = 749, 5 digit numbers

Therefore, the number of 5-digit numbers divisible by either 60 or 70 but are not divisible by 120 since any 5-digit number divisible by 120 is divisible by 60 = 2784 - 749 = 2035, 5-digit numbers.