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Did you solve it? Sexy maths Did you solve it? Sexy maths
(about 16 hours later)
The answers to today’s prime teasersThe answers to today’s prime teasers
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 Leo Woodall (above.)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 Leo Woodall (above.)
Here they are again with solutions.Here they are again with solutions.
1. Prime power1. Prime power
Without using a calculator, what is the last digit of 3 to the power of a million, i.e 31,000,000?Without using a calculator, what is the last digit of 3 to the power of a million, i.e 31,000,000?
Solution: 1Solution: 1
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.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-Eleven2. The 7-Eleven
A shopper walks into a convenience store and buys four items. The cashier says the total is £7.11.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.’‘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.‘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.‘Aren’t you supposed to add the items together?’ replies the shopper.
‘You can do if you like – the total is exactly the same.’‘You can do if you like – the total is exactly the same.’
What are the prices of the items?What are the prices of the items?
Solution £3.16, £1.25, £1.20 and £1.50Solution £3.16, £1.25, £1.20 and £1.50
Let the prices be a, b, c and d. Then we knowLet the prices be a, b, c and d. Then we know
[1] a + b + c + d = 7.11[1] a + b + c + d = 7.11
[2] a x b x c x d = abcd = 7.11[2] a x b x c x d = abcd = 7.11
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.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 = 711[3] 100a + 100b + 100c + 100d = 711
[4] 100a x 100b x 100c x 100d = 100,000,000abcd = 711,000,000[4] 100a x 100b x 100c x 100d = 100,000,000abcd = 711,000,000
To keep things neat, let A = 100a, B = 100b, etc, so we can rephrase the above.To keep things neat, let A = 100a, B = 100b, etc, so we can rephrase the above.
[3] A + B + C + D = 711[3] A + B + C + D = 711
[4] ABCD = 711,000,000[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 where I set the puzzle I explained that all numbers can be broken down into their constituent prime numbers.)We need to solve these two equations, and we do this by first finding the prime factors of 711 million. (In the post where I set the puzzle I explained that all numbers can be broken down into their constituent prime numbers.)
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 to find prime factors)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 to find prime factors)
Now we work out which of these prime factors make up A, B, C and D.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.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. The hint also said that the other three prices divide by 5p. Thus we know that 5 is a factor of B, C 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.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.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.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+.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.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.