Question

How many natural number are there less than 1000 which are divisible by 3 and 5 but not by 35

Answer 1

Answer:-

• The sequence of natural numbers less than 1000 which are divisible by 3 is 3 , 6 , 9 , ... 999.

• The sequence of natural numbers less than 1000 which are divisible by 5 is 5 , 10 , 15 , 20 , 25 ... 995.

• The sequence with the common elements from both is 15 , 30 , ... 990.

If we assume that the sequence is in AP ,

a = 15

d = 30 - 15 = 15

aⁿ = 990

We know that,

aⁿ = a + (n - 1)d

990 = 15 + (n - 1)15

990 = 15 + 15n - 15

990 = 15n

n = 990/15

n = 66.

• The sequence of natural numbers less than 1000 which are divisible by 35 is 35 , 70 , 105 , 140....980.

Similarly,

a = 35

d = 70 - 35 = 35

aⁿ = 980

980 = 35 + (n - 1)35

980 = 35 + 35n - 35

980 = 35n

n = 980/35

n = 28

Numbers which are divisible by both 3 & 5 but not by 35 = 66 - 28 = 38

Therefore, there are 38 numbers which are divisible by both 3 & 5 but not by 35.

Answer 2

Answer:

$\huge\frak{\fcolorbox{white}{pink}{Answer:–}}$

$\rm→ \frac{1000}{3} \frac{1000}{4} \frac{1000}{12}$

$\rm→333+ 200- 83 \\ \rm→250 + 200=450-1 \\ \rm = 449$