1-

Take the number 192 and multiply it by each of 1, 2, and 3:

192 1 = 192
192 2 = 384
192 3 = 576

By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)

The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).

What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, ... , n) where n 1?

 

2-

The first two consecutive numbers to have two distinct prime factors are:

14 = 2 7
15 = 3 5

The first three consecutive numbers to have three distinct prime factors are:

644 = 2² 7 23
645 = 3 5 43
646 = 2 17 19.

Find the first four consecutive integers to have four distinct primes factors. What is the first of these numbers?

 

 

3-

The series, 1¹ + 2² + 3³ + ... + 10¹⁰ = 10405071317.

Find the last ten digits of the series, 1¹ + 2² + 3³ + ... + 1000¹⁰⁰⁰.

 

4-

There are exactly ten ways of selecting three from five, 12345:

123, 124, 125, 134, 135, 145, 234, 235, 245, and 345

In combinatorics, we use the notation, ⁵C₃ = 10.

In general,

ⁿC_r =

n!

r!(nr)!

,where r n, n! = n(n1)...321, and 0! = 1.

It is not until n = 23, that a value exceeds one-million: ²³C₁₀ = 1144066.

How many, not necessarily distinct, values of  ⁿC_r, for 1 n 100, are greater than one-million?

 

5-

A googol (10¹⁰⁰) is a massive number: one followed by one-hundred zeros; 100¹⁰⁰ is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1.

Considering natural numbers of the form, a^b, where a, b 100, what is the maximum digital sum?

 

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