Did you solve it? Sexy maths

Earlier today I set two puzzles involving prime numbers – in particular 3, 7 and 11. Primes are on this month’s cultural radar thanks to the Apple TV+ thriller Prime Target, which stars hunk

du jour

Here they are again with solutions.

1. Prime power

Without using a calculator, what is the last digit of 3 to the power of a million, i.e 3

Solution

When you multiply 3 by itself, the final digit of the answer cycles through the digits 3, 9, 7 and 1. Since a million is divisible by four, the final digit of 3 to the million is 1.

2. The 7-Eleven

A shopper walks into a convenience store and buys four items. The cashier says the total is £7.11.

‘

How funny,’ says the shopper. ‘That’s the name of the shop.’

‘

What’s also funny is that I multiplied the prices of the four items together,’ says the cashier.

‘Aren’t you supposed to add the items together?’ replies the shopper.

‘You can do if you like – the total is exactly the same.’

What are the prices of the items?

Solution £

Let the prices be

a, b, c

[1]

a + b + c + d

[2]

a x b x c x d = abcd =

We can assume that the prices in the shop are in pounds and pence, so lets multiply each of them by 100 to get rid of the decimal point.

[3]

100a + 100b + 100c + 100d

[4] 100

a x

To keep things neat, let A = 100

a

[3] A + B + C + D = 711

[4] ABCD = 711,000,000

We need to solve these two equations, and we do this by first finding the prime factors of 711 million. (

In the post

The prime factorisation of 711 million is 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5 x 79. ( I said you were allowed to

use this site

Now we work out which of these prime factors make up A, B, C and D.

You do this through trial and error, which can be time consuming so at noon I gave the hint that £3.16 was a solution. This corresponds to 316 (since we multiplied the prices by 100). The prime factors of 316 are 79 x 2 x 2. So lets say A = 79 x 2 x 2.

The hint also said that the other three prices divide by 5p. Thus we know that 5 is a factor of B, and D. We also know from [3] that B + C + D = 395.

By playing around with the prime numbers in our factorisation, we get B = 5 x 5 x 5 = 125, C = 5 x 3 x 2 x 2 = 120, and D = 5 x 5 x 3 x 2 = 150. Thus the prices are £1.25, £1.20 and £1.50. Check them if you dont believe me! They add up and multiply to £7.11.

What’s great about this puzzle not so much the solving of it, but the fact that it is amusing to find such an arithmetical gem in a global brand.

I hope you enjoyed the puzzles, I’ll be back in two weeks.

Thanks to Steve Thompson who suggested the first puzzle, and to the PR team at Apple TV+.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one,

email me

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