Ratio, Proportion, Indices, LogarithmMCQMTP Sep 24 Series IIQuestion 928 of 302
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The number of prime factors in 620×358×(15)10\displaystyle 6^{20} ×\displaystyle \times 35^8 ×\displaystyle \times (15)^{10} is:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option d — None of these

All Options:

  • A112
  • B66
  • C56
  • DNone of these

Detailed Solution & Explanation

We are given the expression:
E=620×358×(15)10E = 6^{20} ×\displaystyle \times 35^8 ×\displaystyle \times (15)^{10}

Let us first decompose each base into its prime factors:
1) 6=2×3\displaystyle 6 = 2 ×\displaystyle \times 3
2) $35 = 5 ×\displaystyle \times 7\displaystyle <br>3) &#36;15 = 3 ×\displaystyle \times 5

Now, substitute these prime factorizations into the expression:
E=(2×3)20×(5×7)8×(3×5)10E = (2 ×\displaystyle \times 3)^{20} ×\displaystyle \times (5 ×\displaystyle \times 7)^8 ×\displaystyle \times (3 ×\displaystyle \times 5)^{10}
Applying the exponent law (ab)p=apbp\displaystyle (ab)^p = a^p b^p:
E=220×320×58×78×310×510E = 2^{20} ×\displaystyle \times 3^{20} ×\displaystyle \times 5^8 ×\displaystyle \times 7^8 ×\displaystyle \times 3^{10} ×\displaystyle \times 5^{10}

Now group and combine the powers of identical prime bases:
E=220×320+10×58+10×78E = 2^{20} ×\displaystyle \times 3^{20 + 10} ×\displaystyle \times 5^{8 + 10} ×\displaystyle \times 7^8
E=220×330×518×78E = 2^{20} ×\displaystyle \times 3^{30} ×\displaystyle \times 5^{18} ×\displaystyle \times 7^8

To find the total number of prime factors (including their multiplicities), we sum the exponents of all the prime bases in the prime factorization:
Total Prime Factors=20+30+18+8=76Total Prime Factors=20+30+18+8=76\displaystyle \text{Total Prime Factors} = 20 + 30 + 18 + 8 = 76

Since $76$ is not among the choices (A: 112, B: 66, C: 56), the correct option is "None of these".

Hence, **Option D** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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