### NUMBERS

1. Explanation :

8888 + 888 + 88 + 8 = 9872

1. Explanation :

Let x - 1936248 = 1635773. Then, x = 1635773 + 1936248 = 3572021

1. Explanation :

We may analyse the given equation as shown : Clearly, 2 + P + R + Q = ll. So, the maximum value of Q can be (11 - 2) i.e., 9 (when P = 0, R = 0);


1 2
+ 5 P 9
+ 3 R 7
+ 2 Q 8
---------
= 1 11 4    

1. Explanation :

5793405 x 9999 = 5793405(10000-1) = 57934050000-5793405 = 57928256595.b

1. Explanation :

986 x 137 + 986 x 863 = 986 x (137 + 863) = 986 x 1000 = 986000.

1. Explanation :

1605 x 1605 = $$(1605)^2$$= $$(1600 + 5)^2$$= $$(1600)^2 + (5)^2$$ + 2 x 1600 x 5 = 2560000 + 25 + 16000 = 2576025

1. Explanation :

$$(a^2 + b^2)$$ = $$\sqrt{(a+b)^2+(a-b)^2}$$....formula
$$(313)^2 + (287)^2$$= $$\sqrt{(313 + 287)^2 + (313 - 287)^2}$$ = $$\sqrt{(600)2 + (26)2}$$ = $$\sqrt{(360000 + 676)}$$ = 180338.

1. Explanation :

Clearly, 16 > Ö241. Prime numbers less than 16 are 2, 3, 5, 7, 11, 13. 241 is not divisible by any one of them. 241 is a prime number

1. Explanation :

Clearly, unit's digit in the given product = unit's digit in $$7^{163} * 1^{72}.$$
Now, 74 gives unit digit 1. $$7^{162}$$ gives unit digit 1, =$$7^{163}$$ gives unit digit (l x 7) = 7. Also, $$1^{72}$$gives unit digit 1. Hence, unit's digit in the product = (7 x 1) = 7.

1. Explanation :

Required unit's digit = unit's digit in $$(4)^{102} + (4)^{103}$$. Now,$$4^2$$ gives unit digit 6 =$$(4)^{102}$$ gives unjt digit 6 =$$(4)^103$$ gives unit digit of the product (6 x 4) i.e., 4. Hence, unit's digit in $$(264)^m + (264)^{103}$$ = unit's digit in (6 + 4) = 0.

1. Explanation :

$$(4)^{11} * (7)^5 * (11)^2$$= $$(2*2)^{11} *(7)^5 *(11)^2$$ = $$2^{11} * 2^{11} * 7^5 * 11^2$$ = $$2^{22} * 7^5 * 11^2$$ Total number of prime factors = (22 + 5 + 2) = 29.

1. Explanation :

Given exp = $$(896)^2 - (204)^2$$ = (896 + 204) (896 - 204) = 1100 * 692 = 761200.

1. Explanation :

Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3.

1. Explanation :

Let the missing digit be x. Sum of digits = (1 + 9 + 7 + x + 5 + 4 + 6 +2) = (34 + x). For (34 + x) to be divisible by 9, x must be replaced by 2 . Hence, the digit in place of * must be 2.

1. Explanation :

The number formed by the last two digits in the given number is 72, which is divisible by 4. Hence, 618703572 is divisible by 4.

1. Explanation :

Since the given number is divisible by 5, so 0 or 5 must come in place of $. But, a number ending with 5 is never divisible by 8. So, 0 will replace$. Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4. Hence, digits in place of * and \$ are 4 and 0 respectively.

1. Explanation :

(Sum of digits at odd places) - (Sum of digits at even places) = (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11. Hence, 4832718 is divisible by 11.

1. Explanation :

24 = 3 x 8, where 3 and 8 are co-primes. The sum of the digits in the 52563744 number is 36, which is divisible by 3. So, the given number is divisible by 3. The number formed by the last 3 digits of the given number is 744, which is divisible by 8. So, the given number is divisible by 8. Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-primes. So, it is divisible by 3 x 8, i.e., 24.

1. Explanation :

On dividing 3000 by 19, we get 17 as remainder. Number to be added = (19 - 17) = 2.

1. Explanation :

On dividing 2000 by 17, we get 11 as remainder. =Required number to be subtracted = 11.

1. Explanation :

On dividing 3105 by 21, we get 18 as remainder. Number to be added to 3105 = (21 - 18) = 3. Hence, required number = 3105 + 3 = 3108.

1. Explanation :

Smallest number of 6 digits is 100000. On dividing 100000 by 111, we get 100 as remainder. Number to be added = (111 - 100) - 11. Hence, required number = 100011.

1. Explanation :

Divisor = (Dividend - Remainder)/ Quotient= (15968-37)/89 =179

1. Explanation :

On dividing the given number by 342, let k be the quotient and 47 as remainder. Then, number – 342k + 47 = (19 x 18k + 19 x 2 + 9) = 19 (18k + 2) + 9. The given number when divided by 19, gives (18k + 2) as quotient and 9 as remainder.

1. Explanation :

1. Explanation :

$$2^{10}$$ = 1024. Unit digit of $$2^{10}$$ x $$2^{10}$$ x $$2^{10}$$ is 4 [as 4 x 4 x 4 gives unit digit 4].Unit digit of 231 is 8.Now, 8 when divided by 5, gives 3 as remainder. Hence, 231 when divided by 5, gives 3 as remainder.

1. Explanation :

The required numbers are 14, 21, 28, 35, 77, 84. This is an A.P. with a = 14 and d = (21 - 14) = 7. Let it contain n terms.Then, Tn = 84 => a + (n - 1) d = 84 => 14 + (n - 1) x 7 = 84 or n = 11. Required number of terms = 11.

1. Explanation :

The given numbers are 1, 3, 5, 7, , 99. This is an A.P. with a = 1 and d = 2. Let it contain n terms. Then, 1 + (n - 1) * 2 = 99 or n = 50. Required sum = $$\frac{n *(first term + last term)}{2}$$ = $$\frac {50 *(1 + 99)}{2}$$ = 2500.

1. Explanation :

All 2 digit numbers divisible by 3 are : 12, 51, 18, 21, , 99. This is an A.P. with a = 12 and d = 3. Let it contain n terms. Then, 12 + (n - 1) x 3 = 99 or n = 30. Required sum = $$\frac {30 * (12+99)}{2}$$ = 1665.

1. Explanation :

Clearly 2,4,8,16……..1024 form a GP. With a=2 and r = 4/2 =2. Let the number of terms be n . Then 2 x $$2^{n-1}$$ =1024 or $$2^{n-1}$$ =512 = $$2^9$$.

1. Explanation :

1. Explanation :

1. Explanation :

Let 8597 - x = 7429 - 4358. Then, x = (8597 + 4358) - 7429 = 12955 - 7429 = 5526.

1. Explanation :

839478 x 625 = 839478 x $$5^4$$ = $$\frac {8394780000}{16}$$= 524673750.

1. Explanation :

983 x 207 - 983 x 107 = 983 x (207 - 107) = 983 x 100 = 98300.

1. Explanation :

1398 x 1398 - (1398)2 = (1400 - 2)2= (1400)2 + (2)2 - 2 x 1400 x 2 =1960000 + 4 - 5600 = 1954404.

1. Explanation :

Given exp = $$(387)^2$$+ $$(114)^2$$+ (2 x 387x 114) = $$a^2$$ + $$b^2$$ + 2ab, where a = 387,b=114 = $$(a+b)^2$$ = $$(387 + 114 )^2$$ = $$(501)^2$$ = 251001.

1. Explanation :

Given exp = $$(81)^2$$ + $$(68)^2$$ – 2x 81 x 68 = $$a^2$$ + $$b^2$$ – 2ab,Where a =81,b=68 = $$(a-b)^2$$= $$(81 –68)^2$$ = $$(13)^2$$ = 169.