Question 722997: Identify the number that satisfies all three of the following conditions:
it is a composite between 62 and 72;
the sum of the digits is a prime number; and
it has more than four factors.
Found 2 solutions by josgarithmetic, 119078:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! The range of possibilities is very limited. Some or the possibilities to first consider could be from the set of {62,63,64,65,66,67,68,69,70,71,72}. It is a COMPOSITE number, so eliminate the primes. Look in {62,63,64,65,66,68,69,70,72}.
Sum of the digits is prime! This cuts down the possibilities to {65,70}. 6+5=11 prime, and 7+0=7 prime.
FOUR FACTORS? Cannot be 65 because this one only has 5 and 13 as its factors. The number must be 70 to satisfy all three conditions. 70=7*10 and 70=35*2, but the condition is still not a fit because also 70=14*5. If you were to rely on a very simple elementary way of thinking, you could choose 65 because in the very elementary sense, the factors of 65 are: 1, 5, 13, and 65.
ANSWER: Must be 65.
Answer by 119078(26)  (Show Source):
You can put this solution on YOUR website! List the numbers out from 62 to 72
{62,63,64,65,66,67,68,69,70,71,72}
It needs to be a composite # so take those primes out!
{62,63,64,65,66,68,69,70,72}
Now find a prime in the sum of the numbers
6+2=8 NOPE
6+3=9 NOPE
6+4=10 NOPE
6+5=11 YEP!
6+6=12 NOPE
6+8=14 NOPE
6+9=15 NOPE
7+0=7 YEP!
7+2=9 NOPE
So now down to the last two numbers
{65&70}
finally find the # with MORE than 4 factors. Very key to see that it wants more than four.
65-> 1, 5, 13, 65 This has only four so it can't be it since we need more than four.
70-> 1, 2, 5, 7, 10, 14, 35, 70 This has WAY more than four for sure, it has eight so this by default and with reason is the right answer.
Answer: 70
Make sure to go through the whole thing sometimes even the smallest mistake can get the best of us. Hope this helped
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