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How many positive integers divide evenly into either 144 or 196 or 150?

 Jul 24, 2022
 #1
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\(144 \Rightarrow 2^4 \times 3^2\)

\(196 \Rightarrow 2^2 \times 7^2\)

\(150 \Rightarrow 5^2 \times 2 \times 3\)

 

144 has \(5 \times 3 = 15\) factors. 

 

196 has \(3 \times 3 = 9\) factors, but we need to subtract 3 (1, 2, and 4)

 

150 has \(2 \times 2 \times 3 = 12 \) factors, but we need to subtract 2 (1 and 2)

 

So, there are \(15 + 9 + 12 - 3 - 2 = \color{brown}\boxed{31}\) numbers. 

 Jul 24, 2022
 #2
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(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 24, 25, 28, 30, 36, 48, 49, 50, 72, 75, 98, 144, 150, 196)==29 such divisors

 Jul 24, 2022

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