The number of positive integers that divide (1890) × (130) × (170) and are not divisible by 45 is:
Explanation:
We first need to find all factors of (1890) × (130) × (170) and then remove those factors which are divisible by 45.
(1890) × (130) × (170) = (2 × 33 × 5 × 7) × (13 × 2 × 5) × (17 × 2 × 5)
⇒ 1890 × 130 × 170 = 23 × 33 × 53 × 7 × 13 × 17
∴ Total factors = (3 + 1)(3 + 1)(3 + 1)(1 + 1)(1 + 1)(1 + 1) = 512
Now for factors that are divisible by 45, least power of 3 should be 2 and that of 5 should be 1. Powers of 2, 7, 13 and 17 can be anything.
∴ Total factors which are divisible by 45 = 4 × 2 × 3 × 2 × 2 × 2 = 192
⇒ Number of factors which are not divisible by 45 = 512 - 192 = 320.
Hence, 320.
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