Solved on Dec 07, 2023

Find the largest value of nn such that 5n5^{n} divides 1×2×3××151 \times 2 \times 3 \times \cdots \times 15.

STEP 1

Assumptions
1. We are looking for the highest power of 55, denoted as 5n5^n, that divides the product 1×2×3××151 \times 2 \times 3 \times \ldots \times 15 without leaving a remainder.
2. We are to consider only the numbers from 11 to 1515 inclusive.

STEP 2

Identify all multiples of 55 in the given range 11 to 1515.
Multiplesof5:5,10,15Multiples\, of\, 5: 5, 10, 15

STEP 3

Determine the power of 55 in each multiple of 55.
- 55 is 515^1. - 1010 is 2×512 \times 5^1. - 1515 is 3×513 \times 5^1.

STEP 4

Count the total number of 55s present in the factors.
- There is one 55 in 55. - There is one 55 in 1010. - There is one 55 in 1515.

STEP 5

Add the powers of 55 from each multiple of 55.
Totalpowersof5=1(from5)+1(from10)+1(from15)Total\, powers\, of\, 5 = 1 (from\, 5) + 1 (from\, 10) + 1 (from\, 15)

STEP 6

Calculate the sum of the powers of 55.
Totalpowersof5=1+1+1=3Total\, powers\, of\, 5 = 1 + 1 + 1 = 3

STEP 7

Conclude that the largest value of nn such that 5n5^n is a factor of the product 1×2××151 \times 2 \times \ldots \times 15 is 33.
The largest value of nn is 33.

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