86-7776 NVS 2019

Duration: 1 min

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The video presents a mathematical problem asking for the total number of factors of 2400^7. The instructor begins by writing the expression 2400^7 on the board. The core of the solution involves prime factorization. The instructor factors 2400 into its prime components, writing 2400 = 2^5 × 3^1 × 5^2. This is then used to express 2400^7 as (2^5 × 3^1 × 5^2)^7, which simplifies to 2^(5×7) × 3^(1×7) × 5^(2×7), or 2^35 × 3^7 × 5^14. The formula for the number of factors of a number n = p^a × q^b × r^c is (a+1)(b+1)(c+1). Applying this, the number of factors is (35+1)(7+1)(14+1) = 36 × 8 × 15. The instructor then multiplies these values step-by-step: 36 × 8 = 288, and 288 × 15 = 4320. The final answer is 4320, which is not among the multiple-choice options (A-30, B-60, C-124, D-144), indicating a potential error in the problem or options.

Chapters

  1. 0:00 – 1:13 00:00-01:13

    The video starts with a question on the screen: 'The total number of factors of 2400^7 is equal to:'. The instructor begins the solution by writing '2400^7' on the board. He then proceeds to factor 2400, writing '2400 = 2^5 × 3^1 × 5^2' on the board. He then applies the exponent of 7 to each prime factor, writing '2400^7 = (2^5 × 3^1 × 5^2)^7 = 2^35 × 3^7 × 5^14'. He then applies the formula for the number of factors, writing '(35+1)(7+1)(14+1)'. He calculates 36 × 8 = 288, and then 288 × 15 = 4320. The final answer is 4320, which is not listed in the options A-30, B-60, C-124, D-144.

The video demonstrates a standard method for finding the number of factors of a large number by first performing prime factorization and then applying the formula (a+1)(b+1)(c+1). The instructor correctly applies the rules of exponents and the factor counting formula. However, the final calculated answer of 4320 does not match any of the provided multiple-choice options, suggesting a discrepancy between the problem and the answer choices.