Earlier today I set two puzzles involving major numbers – in particular 3, 7 and 11. The primes are on this month’s cultural radar thanks to the target of Apple TV+ Thriller Prime, which are the Star hunk Du jour Leo Woodall (above.)
They are here again with solutions.
1. Prime Power
Without using a calculator, what is the last digit of 3 to the power of one million, ie 31,000,000?
Solution: 1
When you multiply 3 on its own, the final digit of the answer cycles by numbers 3, 9, 7 and 1. Since one million is divided into four, the final number 3 to million is 1 .
2. The 7-Eleven
A buyer walks into a comfort store and buy four items. The cashier said the total was £ 7.11.
‘How funny, ‘said the consumer. ‘That’s the name of the shop.’
‘The funny thing is that I multiply the prices of four items together, ‘said the cabinet.
‘Shouldn’t you add items?’ Buyer’s response.
‘You can do it if you like – the whole is exactly the same.’
What are the prices of items?
Solution £3.16, £ 1.25, £ 1.20 and £ 1.50
Let the prices a, b, c and d. Then we know
[1] A + B + C + D. = 7.11
[2] axbxcxd = abcd = 7.11
We can imagine that the prices in the shop are in pounds and pence, so each of them is allowed by 100 to get rid of the decimal point.
[3] 100A + 100B + 100c + 100D = 711
[4] 100Ax 100Bx 100Cx 100D = 100,000,000ABCD = 711,000,000
To keep things fixed, let a = 100aB = 100b, etc., so we can redefine the above.
[3] A + B + C + D. = 711
[4] ABCD = 711,000,000
We need to solve these two equations, and we do this by first searching the main factors of 711 million. (In the post where I set the puzzle I explained that all the numbers could break into their covered basic numbers.)
The main 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 are allowed to use this site to find a prime factors)
Today we are working on which of these major factors that 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 indication that £ 3.16 is a solution. This corresponds to 316 (because we multiply 100 prices). The main factors of 316 are 79 x 2 x 2. So say a = 79 x 2 x 2.
The hint also said that the other three prices divide 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 the main 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. So the prices are £ 1.25, £ 1.20 and £ 1.50. Check them if you don’t believe me! They added and increased to £ 7.11.
What is great about this puzzle is not much resolving, but the fact that it is entertaining to find such an arithmetic gem in a global brand.
I hope you enjoy the puzzles, I’ll be back in two weeks.
Thanks to Steve Thompson who suggested the first puzzle, and the PR team on Apple TV+.
I’ve set a puzzle here on alternate Monday since 2015. I’ve always been out for great puzzles. If you want to suggest one, email me.
