Aptitude Discussion

Q. |
How many zeroes will be there in the expansion of the expression $1^1 × 2^2 × 3^3 × 4^4……..× 100^{100}$ |

✖ A. |
1200 |

✖ B. |
1232 |

✔ C. |
1300 |

✖ D. |
1296 |

**Solution:**

Option(**C**) is correct

Number of zeroes will be decided by the power of 2 and 5 in the product.

Since, the power of 5 is less than the power of 2 hence number of zero will be equal to power of 5.

Power of $5 = 5^5 × 10^{10} × 15^{15} × …… × 100^{100}$

⇒ $(5+10+15+20+ (25×2)+ 30+40+.....)$

⇒ $(5+10+15+....100) + (25+50+75+100)$

⇒ $\dfrac{20}{2} [2×5 + 19×5] + 250$

⇒ $1050+250 =$ **1300**

**AKSHAY SATIJA**

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**Preeta**

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Why there is $25 \times 2$ in the sum???

25 is square value of 5. Which means we have $5^{2 \times 25}$ for $25^{25}$

**Shreya**

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Can anyone please give me some simpler explanation...

Any other method to solve it ?