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BartSMP [9]
4 years ago
12

How many numbers between 24 and 719 are divisible by 4 and 8?

Mathematics
1 answer:
Alex Ar [27]4 years ago
3 0

There total 92 number which is divisible by 4 and 8 ,which is below

24,32, 40, 48,56, 64, 72, 80, 88, 96, 104,112,120, 128, 136, 144, 152, 160,168, 176, 184,192, 200,208,216, 224, 232,240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424,432, 440, 448,456,464, 472, 480, 488, 496, 504, 512,520, 528, 536, 544, 552,560, 568, 576, 584, 592, 600, 608, 616,624, 632, 640,648, 656, 664, 672,680, 688, 696, 704, 712,720,728

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In a GP, the sixth term is eight times the tenth term. The sum of the seventh and the eighth term is 192. Find the common ratio
liubo4ka [24]

Let <em>a</em>(<em>n</em>) denote the <em>n</em>-th term in the GP. We're given

<em>a</em>(6) = 8<em>a</em>(10)

<em>a</em>(7) + <em>a</em>(8) = 192

In a geometric progression, consecutive terms are scaled up or down by a fixed ratio <em>r</em> such that

<em>a</em>(<em>n</em>) = <em>r</em> <em>a</em>(<em>n</em> - 1)

By substitution, we get

<em>a</em>(<em>n</em>) = <em>r</em> (<em>r</em> <em>a</em>(<em>n</em> - 2)) = <em>r</em> ² <em>a</em>(<em>n</em> - 2)

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<em>a</em>(<em>n</em>) = <em>r</em> <em>a</em>(<em>n</em> - 1) = <em>r</em> ² <em>a</em>(<em>n</em> - 2) = <em>r</em> ³ <em>a</em>(<em>n</em> - 3) = … = <em>rⁿ </em>⁻¹ <em>a</em>(1)

So we can rewrite the first two equations in terms of the first term of the GP <em>a</em>(1) and the common ratio <em>r</em> :

<em>r</em> ⁵ <em>a</em>(1) = 8<em>r</em> ⁹ <em>a</em>(1)

<em>r</em> ⁶ <em>a</em>(1) + <em>r</em> ⁷ <em>a</em>(1) = 192

Solve the first equation for <em>r</em> :

<em>r</em> ⁵ <em>a</em>(1) = 8<em>r</em> ⁹ <em>a</em>(1)   →   1 = 8<em>r</em> ⁴   →   <em>r</em> = 1/∜8 ≈ 0.5946

Solve the second equation for <em>a</em>(1) :

<em>r</em> ⁶ <em>a</em>(1) + <em>r</em> ⁷ <em>a</em>(1) = <em>r</em> ⁶ <em>a</em>(1) (1 + <em>r</em> ) = 1/(16√2) <em>a</em>(1) (1 + 1/∜8) = 192

→   <em>a</em>(1) = 6144 ∜2 / (1 + ∜8) ≈ 2724.48

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