Question

How many natural number im 100 and 500 whuch are multiples of 3 and 5?

Answer 1

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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

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