Question

424^111×727^188×828^199 find the unit digit​

Answer 1

Answer:

Step-by-step explanation:

Lets multiply the given ones digits so,

First4*1=4. (424¹¹¹)

7*8=56. (727*828)

8*9=72. (828*199)

At the last step multiply each ones place digits

Then,

4*6*2=48

Ones digit is 8

If you understood pls like

Answer 2

Consider 424¹¹¹.  

Ones digit of 424 is 4, which is equal to the remainder got on dividing it by 10. Thus,  

424 ≡ 4 (mod 10)  

Squaring both sides,  

424² ≡ 4² (mod 10)

⇒ 424² ≡ 16 (mod 10)

As 16 ≡ 6 (mod 10),  

⇒ 424² ≡ 6 (mod 10)  

The remainder got here is 6. So the same remainder 6 is also got on dividing the powers of 424² by 10.  

⇒ (424²)ⁿ ≡ 6 (mod 10)  

So,  

(424²)⁵⁵ ≡ 6 (mod 10)

⇒ 424¹¹⁰ ≡ 6 (mod 10)

Now, multiply both sides by 424.  

⇒ 424¹¹⁰ × 424 ≡ 6 × 424 (mod 10)  

But, according to 424 ≡ 4 (mod 10),  

⇒ 424¹¹⁰ × 424 ≡ 6 × 4 (mod 10)

⇒ 424¹¹¹ ≡ 24 (mod 10)

⇒ 424¹¹¹ ≡ 4 (mod 10)        [∵ 24 ≡ 4 (mod 10)]

Thus the ones digit of 424¹¹¹ is 4.

Consider 727¹⁸⁸.

Ones digit of 727 is 7.

⇒ 727 ≡ 7 (mod 10)

727⁴ ≡ 7⁴ (mod 10)

⇒ 727⁴ ≡ 2401 (mod 10)

⇒ 727⁴ ≡ 1 (mod 10)          [∵ 2401 ≡ 1 (mod 10)]

(727⁴)⁴⁷ ≡ 1⁴⁷ (mod 10)

⇒ 727¹⁸⁸ ≡ 1 (mod 10)

Thus the ones digit of 727¹⁸⁸ is 1.

Consider 828¹⁹⁹.

Ones digit of 828 is 8.

⇒ 828 ≡ 8 (mod 10)

828³ ≡ 8³ (mod 10)

⇒ 828³ ≡ 512 (mod 10)

⇒ 828³ ≡ 2 (mod 10)          →   (1)          [∵ 512 ≡ 2 (mod 10)]          

828⁴ ≡ 8⁴ (mod 10)

⇒ 828⁴ ≡ 4096 (mod 10)

⇒ 828⁴ ≡ 6 (mod 10)          [∵ 4096 ≡ 6 (mod 10)]

∴ (828⁴)ⁿ ≡ 6 (mod 10)

(828⁴)⁴⁹ ≡ 6 (mod 10)

⇒ 828¹⁹⁶ ≡ 6 (mod 10)

828¹⁹⁶ × 828³ ≡ 6 × 828³ (mod 10)

⇒ 828¹⁹⁹ ≡ 6 × 2 (mod 10)          [From (1)]

⇒ 828¹⁹⁹ ≡ 12 (mod 10)

⇒ 828¹⁹⁹ ≡ 2 (mod 10)          [∵ 12 ≡ 2 (mod 10)]

Thus the ones digit of 828¹⁹⁹ is 2.

Now multiply each ones digits.

⇒ 424¹¹¹ × 727¹⁸⁸ × 828¹⁹⁹ ≡ 4 × 1 × 2 (mod 10)

⇒ 424¹¹¹ × 727¹⁸⁸ × 828¹⁹⁹ ≡ 8 (mod 10)

Thus the answer is 8.