How-many-different-three-digit-numbers-a-satisfy-the-condition-GCD-a-18-2-

Question Number 202109 by hardmath last updated on 20/Dec/23

Answered by mr W last updated on 22/Dec/23

100 ⩽ a ⩽ 999

18 = 2 \times 3^{2}

18 h a s f o l l o w i n g f a c t o r s w h i c h a r e

e q u a l o r g r e a t e r t h a n 2 :

2, 3, 6, 9, 18

t h a t m e a n s a m u s t b e a m u l t i p l e o f

2 o r 3 o r 6 o r 9 o r 18 .

a = 2 k : \Rightarrow ⌊ \frac{999}{2} ⌋ - ⌊ \frac{99}{2} ⌋ = 499 - 49 = 450

a = 3 k : \Rightarrow ⌊ \frac{999}{3} ⌋ - ⌊ \frac{99}{3} ⌋ = 333 - 33 = 300

a = 6 k : \Rightarrow ⌊ \frac{999}{6} ⌋ - ⌊ \frac{99}{6} ⌋ = 166 - 16 = 150

a = 9 k : \Rightarrow ⌊ \frac{999}{9} ⌋ - ⌊ \frac{99}{9} ⌋ = 111 - 11 = 100

a = 18 k : \Rightarrow ⌊ \frac{999}{18} ⌋ - ⌊ \frac{99}{18} ⌋ = 55 - 5 = 50

18 k 9 k 6 k 3 k 2 k

50 + 100 + 150 + 300 + 450

50 + \cancel \underset{50}{100} + \cancel \underset{100}{150} + \cancel \underset{\cancel \underset{\cancel \underset{100}{200}}{250}}{300} + \cancel \underset{\cancel \underset{}{400}}{450}

50 + 50 + 100 + 100 + 300 = 600 n u m b e r s ✓

Commented by hardmath last updated on 21/Dec/23

perfect dear professor, thank you so much

Commented by hardmath last updated on 21/Dec/23

My dear professor, excuse me, what does the red part on the last line mean? And where did they come from?

Commented by mr W last updated on 22/Dec/23

i f a n u m b e r i s d i v i s i b l e b y 18, t h e n

i t i s a l s o d i v i s i b l e b y 9, b y 2, b y 6, b y 3 .

s i m i l a r l y i f a n u m b e r i s d i v i s i b l e

b y 9, t h e n i t i s a l s o d i v i s i b l e b y 3 .

a n d i f i t i s d i v i s i b l e b y 6, i t i s a l s o

d i v i s i b l e b y 3 a n d b y 2 .

s o w e m u s t s u b s t r a c t t h e n u m b e r s

w h i c h a r e d o u b l y c o u n t e d a s s h o w n

i n t h e l a s t l i n e .

Commented by hardmath last updated on 22/Dec/23

Excellent dear pfofessor,

thank you very mych