Question

Find the number of zeroes in 2^2×5^4×4^6×10^8×6^12×15^14.​

Answer 1

Answer:

Step 1 : Suppose you have to find the number of zeroes in a product: 24 × 32 × 17 × 23 × 19. We first get the series in terms of its prime factors i.e. 23∗31∗25∗171∗19∗23. As you can notice, this product will have no zeroes because it has no 5 in it.

However, if you have an expression like: 8 × 15 × 23 × 17 × 25 × 22 The above expression can be rewritten in the standard form as: 23∗31∗51∗23∗17∗52∗21∗111

Step 2 : Zeroes are formed by a combination of 2 × 5. Hence, the number of zeroes will depend on the number of pairs of 2’s and 5’s that can be formed. In the above product, there are four twos and three fives. Hence, we shall be able to form only three pairs of (2 × 5). Hence, there will be 3 zeroes in the product.

Finding the Number of Zeroes in a Factorial Value :

Suppose you had to find the number of zeroes in 6!. 6! = 6 × 5 × 4 × 3 × 2 × 1 = (3 × 2) × (5) × (2 × 2) × (3) × (2) × (1). Counting the number of 5 will give the answer.

Method 2 : For finding the zeroes in 6! we use 65+653+... So we get 1 as the answer as all divisions after the first term in the series are in decimals which we ignore.