(16^3 + 17^3+ 18^3+ 19^3), divided by 70 leaves a remainder ?

Answer 1

False. (16^3 + 17^3+ 18^3+ 19^3)/70=310, with no remainder.

Answer 2

There's no remainder, and here's how you can prove it without using a calculator *or* even bothering to work out all the numbers. It depends on knowing this key factoring pattern, commonly taught in algebra classes: for any pair of numbers a and b,

a³ + b³ = (a + b)(a² - ab + b²)

So we start by recognizing that 70 = 35·2. Next, we rearrange the terms in the problem to look like this:

16³ + 19³ + 17³ + 18³

We can group the terms in pairs like this:

(16³ + 19³) + (17³ + 18³)

Now we factor each pair, using the pattern above:

(16 + 19)(16² - 16·19 + 19²) + (17 + 18)(17² - 17·18 + 18²)

now, 16 + 19 and 17 + 18 both equal 35, so:

(35)(16² - 16·19 + 19²) + (35)(17² - 17·18 + 18²)

We can now factor out the 35:

(35)(16² - 16·19 + 19² + 17² - 17·18 + 18²)

Now, let's look at that long expression in the parentheses. It's the sum of four even numbers (16², 16·19, 17·18, and 18²) and two odd numbers (19² and 17²). Since any two odd numbers always add up to an even number, the terms in the parentheses all add up to some even number. We don't know what it is, but we don't care -- all we care is that it's even, which means it contains a factor of 2.

So our number has a factor of 2 and a factor of 35 -- and since 70 = 35·2, our number has a factor of 70. Which means that dividing it by 70 leaves no remainder.

I'm betting your teacher will give more credit for an answer like that for one like "the calculator says so." :-)

Hope that helps!

Answer 3

16^3= 16 * 16 * 16 = 4096
17^3= 17 * 17 * 17 = 4913
18^3 = 18 * 18 * 18 = 5832
19^3 = 19 * 19 * 19 = 6859

4096+4913+5832+6859/ 70
= 2170/ 70
= 310
no remainder.
( just use a calculator)

Answer 4

16^3= last digit of this number is 6
17^3= is3
18^3= is 2
19^3= is 9
add all the last digit=6+3+2+9
=20 which is multiple of 2 an even number
when it is divided by 70 then remainder gives 0 ..simple just multiply last digits n add them.


thank you

manish

Answer 5

The remainder is zero.....
(16^3 + 17^3 + 18^3 + 19^3)/70 = 310 --> exactly

Answer 6

Answer is 310 remainder nil

Answer 7

310 no remainder

Answer 8

310 even... no remainder

Answer 9

dont know everyone else says 310 lol so i guess 310

Answer 10

No. The remaining is zero. Use mod function in Excel.