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1. The last digit of the number obtained by multiplying
the numbers 81 x 82 x 83 x 84 x 85 x 86 x 87 x 88 x 89 will be

Correct Answers :

[A]

Explanation :

The units digit in this case would obviously be '0' because the given expression has a pair of 2 and 5 in it's prime factors

2. The sum of the digits of a two-digit number is 10,
while when the digits are reversed, the number decreases by 54. Find the changed number.

Correct Answers :

[A]

Explanation :

When you read the sentence “when the digits are
reversed, the number decreases by 54, you should automatically get two reactions going in your mind.
(i) The difference between the digits would be 54/9 = 6.

(ii) Since the number 'decreases'- the tens digit of the number would be larger than the units digit.

Also, since we know that the sum of the digits is 10, we get that the digits must be 8 and 2 and the
number must be 82. Thus, the changed number is 28.

3. When we multiply a certain two-digit number by the sum of its digits, 405 is achieved. If you multiply the number written in reverse order of the same digits by the sum of the digits, we get 486. Find the number.

Correct Answers :

[B]

Explanation :

The two numbers should be factors of 405. A factor search will yield the factors. (look only for 2 digit factors of 405 with sum of digits between 1 to 19).

Also 405 = 5 x 34 . Hence: 15 x 27, 45 x 9 are the only two options.

From these factors pairs only the second pair gives us the desired result.

i.e. Number x sum of digits = 405.

Hence, the answer is 45.

4. The sum of two numbers is 15 and their geometric
mean is 20% lower than their arithmetic mean. Find the numbers.

Correct Answers :

[B]

Explanation :

You can solve this question by using options. It can be seen that Option (b) 12,3 fits the situation perfectly as their Arithmetic mean = 7.5 and their geometric mean = 6 and the geometric mean is 20% less than the arithmetic mean.

5. The difference between two numbers is 48 and the
difference between the arithmetic mean and the geometric mean is two more than half of 1/3 of 96. Find
the numbers.

Correct Answers :

[A]

Explanation :

Two more than half of 1/3^{rd} of 96 = 18. Also since
we are given that the difference between the AM and GM is 18, it means that the GM must be an
integer. From amongst the options, only option (a) gives us a GM which is an integer. Thus, checking
for option 1, we get the GM = 7 and AM = 25.

6. If *A*381 is divisible by 11, find the value of the
smallest natural number *A*.

Correct Answers :

[C]

Explanation :

of the even placed digits and the odds placed digits should be either 0 or a multiple of 11. This means that (A + 8) – (3 + 1) should be a multiple of 11 – as it is not possible to make it zero. Thus, the smallest value that A can take (and in fact the only value it can take) is 7. Option (c) is correct.

7. If 381*A* is divisible by 9, find the value of smallest
natural number *A*.

Correct Answers :

[D]

Explanation :

For 381*A* to be divisible by 9, the sum of the digits
3 + 8 + 1 + *A* should be divisible by 9. For that to happen *A* should be 6. Option (d) is correct.

8. What will be the remainder obtained when (9^{6} + 1) will be divided by 8?

Correct Answers :

[D]

Explanation :

9^{6} when divided by 8, would give a remainder of 1.
Hence, the required answer would be 2.

9. Find the ratio between the LCM and HCF of 5, 15 and 20.

Correct Answers :

[D]

Explanation :

LCM of 5, 15 and 20 = 60. HCF of 5, 15 and 20 = 5. The required ratio is 60:5 = 12:1

10. Find the LCM of 5/2, 8/9, 11/14.

Correct Answers :

[D]

Explanation :

LCM of 5/2, 8/9 and 11/14 would be given by: (LCM of numerators)/(HCF of denominators) = 440/1 = 440

11. If the number A is even, which of the following will
be true?

Correct Answers :

[A]

Explanation :

Only the first option can be verified to be true in this case. If A is even, 3A would always be divisible by 6 as it would be divisible by both 2 and 3. Options b and c can be seen to be incorrect by assuming the value of A as 4.

12. A five-digit number is taken. Sum of the first four digits (excluding the number at the units digit) equals sum of all the five digits. Which of the following will not divide this number necessarily?

Correct Answers :

[C]

Explanation :

The essence of this question is in the fact that the last digit of the number is 0. Naturally, the number is necessarily divisible by 2,5 and 10. Only 4 does not necessarily divide it.

13. A number 15*B* is divisible by 6. Which of these will
be true about the positive integer *B*?

Correct Answers :

[D]

Explanation :

B would necessarily be even- as the possible values of B for the three digit number 15B to be divisible by 6 are 0 and 6. Also, the condition stated in option (c) is also seen to be true in this case — as both 0 and 6 are divisible by 6. Thus, option (d) is correct.

14. Two numbers *P* = 2^{3} .3^{10}.5 and *Q* = 2^{5} .3^{1}.7^{1} are given.
Find the GCD of *P* and *Q*.

Correct Answers :

[D]

Explanation :

For the GCD take the least powers of all common
prime factors. Thus, the required answer would be 2^{3} x 3

15.Find the units digit of the expression 25^{6251} + 36^{528} + 73^{54}

Correct Answers :

[B]

Explanation :

The units digit would be given by 5 + 6 + 9 (numbers ending in 5 and 6 would always end in 5 and 6 irrespective of the power and 3^{54} will give a units digit equivalent to 3^{4n+2} which would give us a unit digit of 3^{2} i.e.9).

16. Find the units digit of the expression 55^{725} + 73^{5810} + 22^{853}.

Correct Answers :

[C]

Explanation :

The respective units digits for the three parts of the expression would be: 5 + 9 + 2 =16 --> required answer is 6. Option (c) is correct.

17. Find the units digit of the expression 11^{1} +12^{2} +13^{3} +14^{4} +15^{5} +16^{6}

Correct Answers :

[B]

Explanation :

The respective units digits for the six parts of the expression would be: 1 + 4 + 7 + 6 + 5 + 6 = 29 --> required answer is 9. Option (b) is correct.

18. Find the units digit of the expression 11^{1} .12^{2} .13^{3} .14^{4} .15^{5} .16^{6}

Correct Answers :

[D]

Explanation :

The respective units digits for the six parts of the expression would be: 1 x 4 x 7 x 6 x 5 x 6 --> required answer is 0. Option (d) is correct.

19. Find the number of zeroes at the end of 1090!

Correct Answers :

[A]

Explanation :

The number of zeroes would be given by adding the quotients when we successively divide 1090 by 5: 1090/5 + 218/5 + 43/5 + 8/5 = 218 + 43 + 8 + 1 = 270. Option (a) is correct.

20. If 146! is divisible by 5^{n} , then find the maximum value of n.

Correct Answers :

[B]

Explanation :

The number of 5's in 146! can be got by [146/5] + [29/5] + [5/5]= 29+5+1 = 35

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