model 4 finding unit place of a number Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (a)

Expression = $(2137)^754$

Unit’s digit in 2137 = 7

Now, $7^1 = 7, 7^2 = 49, 7^3 = 343, 7^4 = 2401, 7^5$ = 16807, ...

Clearly, after index 4, the unit’s digit follow the same order.

Dividing index 754 by 4 we get remainder = 2

∴ Unit’s digit in the expansion of $(2137)^754$ = Unit’s digit in the expansion of $(2137)^2$ = 9

Answer: (c)

Unit’s digit in the expansion of $(22)^23$

= Unit’s digit in the expansion of $(2)^23$

Now, $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32$

i.e. 2 repeats itself after the index 4.

On dividing 23 by 4, remainder = 3

∴ Unit’s digit in $(2)^23$ = Unit’s digit in $(2)^3$ = 8

Answer: (b)

$(4)^{2m}$ gives 6 at unit digit.

$(4)^{2m +1}$ gives 4 at unit digit.

$(5)^n$ gives 5.

The same is the case with 1.

Required digit = Unit’s digit in the product of 4×5 × 1 = 0

Answer: (d)

Last digit of $(1001)^2008$ + 1002 = 1 + 2 = 3

Answer: (b)

Let the two–digit number be 10x + y where x < y.

Number obtained on reversing the digits =10y + x

According to the question,

10y + x = 4 (10x + y) – 24

40x + 4y – 10y – x = 24

39x – 6y = 24

13x – 2y = 8 ....(i)

Again, y – x = 7

y = x + 7 ....(ii)

13x – 2 (x + 7) = 8

13x – 2x – 14 = 8

11 x = 14 + 8 = 22

x = $22/11$ = 2

From equation (ii),

y – 2 = 7 ⇒ y = 2 + 7 = 9

Number = 10x + y =10×2+9= 29