### Brain Teasers

# Primes by Primes

Consider two-digit prime numbers. Whenever you add the two digits you get a number which is not prime i.e. It may be divisible by 2,3, etc. However there are only five two-digit prime numbers whose digits when added give a result that is divisible by 5. Three are 19,23 and 73.

Can you find the other two prime numbers whose digits add up to a number divisible by 5?

Can you find the other two prime numbers whose digits add up to a number divisible by 5?

### Answer

They are 37 and 41.Hide Answer Show Answer

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## Comments

It is not true that you necessarily

get "a number which is not prime,"

as your example of 23 shows. Another

example is 43. However, no prime will

give you a digit sum divisible by 3 (unless you count

03), as any number with

a digit sum divisible by three must

have been divisible by three itself.

So when looking for the two-digit primes with

a digit sum divisible by 5, it is sufficient

to look at numbers with digit sums of 5 and

10 only. No prime will have a digit sum of 15,

and the maximum digit sum for two-digit numbers

is 18 (from 99).

get "a number which is not prime,"

as your example of 23 shows. Another

example is 43. However, no prime will

give you a digit sum divisible by 3 (unless you count

03), as any number with

a digit sum divisible by three must

have been divisible by three itself.

So when looking for the two-digit primes with

a digit sum divisible by 5, it is sufficient

to look at numbers with digit sums of 5 and

10 only. No prime will have a digit sum of 15,

and the maximum digit sum for two-digit numbers

is 18 (from 99).

and when you add 37 and 41 i dont think that the sum is divisble by 5

I like the puzzle. Good comment dewie. I was thinking about 87 until I realised it was divisible by 3 and hence not prime. Of course you are right. If the sum of the digits were 15 then that would have made it automatically divisible by 3. Good one!

ccccc, even though you commented on that 3 years ago...

You add together the digits, not the numbers.

37 | 3 + 7 | 40 / 5 = 4

41 | 4 + 1 | 5 / 5 = 1

You add together the digits, not the numbers.

37 | 3 + 7 | 40 / 5 = 4

41 | 4 + 1 | 5 / 5 = 1

sane 3+ 7 is 10 not 40

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