Wow. I don't know any fancy way to do this but I would just go up the list until I was able to rule out four of the five numbers. Start with 4 + 9 + 16 +25 + 36 + 49 + 64 then go from there.
Hello MHF,
Which of the following cannot be the last digit of the sum of the squares of seven consecutive numbers?
A: 3
B:5
C:6
D:7
E:8
I have no idea where to begin or what to do, i tried using the squares of 1-7
but obviously that wouldn't give me an answer (hopeless thing to do)
Any help appreciated.
I can only offer a trial and error method:
Consider the last digit of a number and the last digit of it's square:
If you add the squares of numbers with end digits 0 to 6 you'll get 31 that means the end digit 1;
if you add the squares of numbers with end digits 1 to 7 you'll get 40 that means the end digit 0;
if you add the squares of numbers with end digits 2 to 8 you'll get 43 that means the end digit 3;
if you add the squares of numbers with end digits 3 to 9 you'll get 40 that means the end digit 0;
and so on...
Compare your results with the given answers (and then pick the right one!)
If I didn't make a mistake the answer is D.
Hello, 99.95!
Note the last digit of all squares.Which of the following cannot be the last digit
of the sum of the squares of seven consecutive numbers?
. .
. . . .
We see that squares cannot end in 2, 3, 7, or 8.
Let the 7 consecutive numbers be: .
Then we have:
. .
Suppose the sum ends in 7.
. . Then must end in 1.
. . And must end in 7.
But no square ends in 7.
Answer: .
Here is another approach. Let
(a well-known identity).
Then the sum of the squares of the 7 consecutive integers ending in n is
Evaluating for n = 0, 1, 2, ..., 9 modulo 10 will then reveal the possibilities for the last digit of the sum.
[Edit] Soroban posted his answer while I was composing this-- essentially the same approach? Or maybe not.[/edit]
Thanks guys both methods (although similiar) are definately useful and yes the answer is "D" (7). May I ask how you guys go about solving these, is it just some logical approach you take?
Thanks again
By the way, does modulo = remainder?
Eg: 10/4= 2 r 2 or 2 mod (2) ?