Question

1^33+2^33+3^33+......89^33 find unit digit​

Answer 1

Answer:

Brother,

See… the units digit of the entire sum can be obtained by adding the individual unit digits of each term.

Lets start with nos. having 1 in units place. No matter how much power they are raised to, the unit’s digit will be 1.

For 2, it follows a pattern. Power 1 → 2, Power 2 → 4, Power 3 →8, Power 4 → 6. And this pattern continues.

For 3, the pattern is 3, 9, 7, 1.

For 4, it is 4, 6.

For 5, it is always 5.

For 6, it is always 6.

For 7, it is 7, 9, 3, 1.

For 8, it is 8, 4, 2, 6.

For 9, it is 9, 1.

For 0, it is always 0.

Now you got the technique. Find them, add them… There you go. The answer is 5.

MARK AS BRAINLIEST

Answer 2

Answer:

The digit in the unit place of the sum 1³³ + 2³³ + 3³³ + .... +89³³ is 5.

Step-by-step explanation:

Given,

The sum 1³³ + 2³³ + 3³³ + .... +89³³.

To find,

The unit digit of the sum of 1³³ + 2³³ + 3³³ + .... +89³³.

Calculation,

We can write 1³³ + 2³³ + 3³³ + .... +89³³ as:

(1³³ + 89³³) + (2³³ + 88³³) + (3³³ + 87³³) +.....+ (44³³ + 46³³) + 45³³....(1)

(We just grouped the sum as pairs as above)

But we know that:

(x³³ + y³³) = (x + y)(x³² - x³¹y + x³⁰y² -... + y³²)

We use the expression for equation (1):

(1³³ + 89³³) = (1 + 89)(n₁) = 90n₁

(2³³ + 88³³) = 90n₂

.

.

.

(44³³ + 46³³) = 90n₄₄

And the last term that is alone is 45³³.

As we can see from all the paired-up terms we have '0' at its unit place. So, adding them will also result in '0' at its unit place.

So 45³³ will decide the unit place of the sum 1³³ + 2³³ + 3³³ + .... +89³³.

But we know that the unit place of 45³³ = 5

Therefore, the digit in the unit place of the sum 1³³ + 2³³ + 3³³ + .... +89³³ is 5.

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