Question

(5!) + (10!)^10! x (50!)^50! x (100!)^100!
last three digit of this expression​

Answer 1

Keys

• Factors and Multiples

→ If A is a multiple of B, B is a factor of A.

• Factor

→ According to the above, if divided by factor there is no remainder.

Solution

To find the last three digits we can divide by 1000.

There are two cases.

• The number has 1000 as a factor. (Case 1)
• The number doesn't have 1000 as a factor. (Case 2)

Case 1. The number is a multiple of 1000.

For $n!$ to be divisible by $2^3\cdot 5^3$, $n$ should equal at least 15. [1]

The product has $15!$ as a factor.

So $(10!)^{10!}\times (50!)^{50!}\times (100!)^{100!}$ is divisible by 1000.

Case 2. The number is not a multiple of 1000.

$5!=2^3\times 3\times 5$ doesn't have enough power of 5 to be divisible by 1000. So, $5!$ is not divisible by 1000.

Conclusion

We conclude the last three digits are the last digits of $5!=120$.

Why?

[1] The multiples of 2 appear more frequently than 5, and there is no perfect square/cube of 5 below 15. 15 is the third multiple of 5, so the power of 5 in $15!$ is exactly 3.

Answer 2

Keys

Factors and Multiples

→ If A is a multiple of B, B is a factor of A.

Factor

→ According to the above, if divided by factor there is no remainder.

Solution

To find the last three digits we can divide by 1000.

There are two cases.

The number has 1000 as a factor. (Case 1)

The number doesn't have 1000 as a factor.

(Case 2)

Case 1. The number is a multiple of 1000.

For n!n! 5^32 _3 ⋅5 _3

, nn should equal at least 15. [1]

The product has 15!15! as a factor.

(10!) 10! ×(50!) 50! ×(100!) 100! is divisible by 1000.

Case 2. The number is not a multiple of 1000.

5!=2^3\times 3\times 55!=2

3

×3×5 doesn't have enough power of 5 to be divisible by 1000. So, 5!5! is not divisible by 1000.

Conclusion

We conclude the last three digits are the last digits of 5!=1205!=120 .

Why?

[1] The multiples of 2 appear more frequently than 5, and there is no perfect square/cube of 5 below 15. 15 is the third multiple of 5, so the power of 5 in 15!15! is exactly 3.