Every Number is Interesting Trivia Quiz

Answer: Prime numbers

A prime number is an integer that is only divisible by itself and 1.

To test if a number is prime, you need to check if it is divisible by every prime number up to its square root (rounded down). The square root of 409 is about 20.2, so we need to check every prime number up to 20. Since 409 is not divisible by 2, 3, 5, 7, 11, 13, 17 or 19, it is prime.

Answer: 673, 73 and 3 are all prime

Left-truncatable primes are prime numbers where you can keep removing the first digit of the number, and be left with a prime. 673, 73 and 3 are all prime numbers, so 673 is a left-truncatable prime.

With 673, you can extend the sequence further; for instance, 4673 is also a left-truncatable prime because 4673, 673, 73 and 3 are all prime. 34673 is another left-truncatable prime. This sequence can carry on all the way up to 739387864234673, a 15 digit number.

There are only 4260 left-truncatable primes in existence.

Answer: 348

"Prime factorisation of x" just means "the prime numbers that multiply to make x". Each number only has one unique prime factorisation.

It is perhaps easiest to work out the answer by elimination, rather than multiplying 2, 2, 3 and 29 together. 577 and 235 are both odd, so they can't have 2 as a factor; 836 is not a multiple of 3, because the sum of its digits is 17, which is not a multiple of 3.

348 is the third number in a row to have an interesting fact relating to primes. It is the sum of four consecutive primes i.e. 79 + 83 + 89 + 97.

Answer: The sum of its digits (18)

486 is not divisible by 4 (halving 486 gives 243, an odd number), which actually eliminates all three of the incorrect answers (as something divisible by 4 cannot be divisible by 8 either).

Harshad numbers are divisible by the sum of their digits: in this case, 4 + 8 + 6 = 18, and 486/18 = 27.

Answer: 13

The fastest way to answer this is probably by estimation, or trial and error with the four given answers.

n = 13 is the only real solution of the cubic equation given:

455 = n*(n+1)*(n+2)/6
455*6 = n*(n+1)*(n+2)
2730 = n*(n^2 + 3n + 2)
2730 = n^3 + 3n^2 + 2n
n^3 + 3n^2 + 2n - 2730 = 0

which gives the solution n = 13 for anyone with a graphic calculator.

Answer: 5

If a number is divisible by 5, it will end in a 5 or a 0; 588 ends in an 8, so it is not divisible by 5.

588 = 2*2*3*7*7

And it is another Harshad number because 5 + 8 + 8 = 21; 588/21 = 28.

Answer: 439

Square numbers are numbers which can be written as an integer multiplied by itself.

4 = 2*2; 9 = 3*3; 4 + 3 + 9 = 16 = 4*4

There are actually 11 three-digit numbers I could have been describing, but 439 was the only one of those numbers that was included as a possible answer. Out of these 11 numbers, 6 of the numbers themselves were square numbers, and 2 of these 6 square numbers also had a middle digit that was a square number.

The first of those two is 144 (1, 4 and 4 are square; 1 + 4 + 4 = 9, which is square; 144 is square). See if you can work out what the other number is.

Answer: It only has 2 prime factors

955 has the prime factors 5 and 191, making it semiprime. It is also the sum of 6 consecutive square numbers, from 10^2 to 15^2: 100 + 121 + 144 + 169 + 196 + 225 = 955.

Answer: 383

Base 9 has only the digits 0-8, instead of the 0-9 we usually use (in "base 10"). To convert 318 from base 9 to base 10, we need to find x, y and z so that: x*81 + y*9 + z = 318. In this case x = 3, y = 8 and z = 3, so the base 9 number is 383.

The shortcut to the answer, the most astute of you may have noticed, was to realise that because base 9 has fewer digits available than base 10, any base 10 number will look bigger in base 9. (For instance, 90 in base 10 is written as 110 in base 9; normally, 110 > 90.) 383 was the only choice bigger than 318, so it must be the right answer.

318 is also the sum of 12 consecutive prime numbers:

7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 = 318.

Answer: 1 + 276 + 406

1 + 276 + 406 = 683.

A triangular number can be written in the form n*(n+1)/2, but an easier way to think about it is that a triangular number is 1 + 2 + 3 + 4 + ..., with any number of terms. So the 26th triangular number is 1 + 2 + 3 + ... + 25 + 26 = 351.

The gap between triangular numbers increases by one every time. The sequence begins 1, 3, 6, 10, 15, ...; the difference between each two consecutive terms is 2, 3, 4, 5, ...

Triangular numbers were named because they can be thought of as the number of dots in an equilateral triangle: e.g. an equilateral triangle with three rows has 1 dot on the top row, 2 on the second row and 3 on the third, which makes 6 dots in total (so 6 is the third triangular number). Tetrahedral numbers were the focus of question 5, and they were based on the same principle but with a 3D triangular pyramid instead of a 2D triangle.