Question

How many natural numbers are there in between 100 and 500 which are multiple of 3 and 5

Answer 1

multiple of 3 : 102,105,108.........495,498

it's an A.p with c.d = 3
a = 102
an = 498

WKT
an = a + (n-1)d
498 = 102 + (n-1)*3
n-1 = 498-102/3
n = 133 (1)

multiple of 5 : 105,110,115.........490,495
it's an A.p with c.d = 5

a = 105
an = 495

WKT
an = a + (n-1)d
495 = 105 + (n-1)*5
n-1 = 495-105/5
n = 79 (2)


multiple of 15(LCM of 3,5): 105,120,135.......480,495
these are common term of 3,5. we need to subtract these as they are counted twice.
it's an A.p with c.d = 15
a = 105
an = 495

WKT
an = a + (n-1)d
495 = 105 + (n-1)*15
n-1 = 495-105/15
n = 27 (3)

Ans = (1)+(2)-(3) = 133 + 79 - 27 = 185

Answer 2

multiple of 3 : 102,105,108.........495,498

it's an A.p with c.d = 3

a = 102

an = 498

WKT

an = a + (n-1)d

498 = 102 + (n-1)*3

n-1 = 498-102/3

n = 133 (1)

multiple of 5 : 105,110,115.........490,495

it's an A.p with c.d = 5

a = 105

an = 495

WKT

an = a + (n-1)d

495 = 105 + (n-1)*5

n-1 = 495-105/5

n = 79 (2)

multiple of 15(LCM of 3,5): 105,120,135.......480,495

these are common term of 3,5. we need to subtract these as they are counted twice.

it's an A.p with c.d = 15

a = 105

an = 495

WKT

an = a + (n-1)d

495 = 105 + (n-1)*15

n-1 = 495-105/15

n = 27 (3)

Ans = (1)+(2)-(3) = 133 + 79 - 27 = 185