Test Level 1: Number System - 2 Solved MCQs CAT

1420 = 142 × 10 = 2² × 71¹ × 5¹.
Thus, the number of factors of the number would be (2 + 1) (1 + 1) (1 + 1) = 3 × 2 × 2 = 12.
Option (d) is correct.

The sides of the pentagon being 1422, 1737, 2160, 2214 and 2358, the least difference between any two numbers is 54.
Hence, the correct answer will be a factor of 54.
Further, since there are some odd numbers in the list, the answer should be an odd factor of 54.
Hence, check with 27, 9 and 3 in that order. You will get 9 as the HCF.

The HCF of the given numbers is 31 and hence the number of bottles required would be
403/31 + 465/31 + 496/31 = 13 + 15 + 16 = 44.
Option (d) is correct.

The LCM of 16, 18 and 20 is 720. The numbers which would give a remainder of 4, when divided by 16, 18 and 20 would be given by the series: 724, 1444, 2164, 2884 and so on.
Checking each of these numbers for divisibility by 7, it can be seen that 2884 is the least number in the series that is divisible by 7 and hence is the correct answer.
Option (d) is correct.

5 × 7 × 6 = 0. Option (c) is correct.
please learn cyclicity of power
125⁸¹³ unit digit will be 5
553³⁷⁰³   unit digit will be 3703/4  calculated as remainder will be 3 so 3³ will be 27 so unit digit will be 7
4532⁸²⁸  unit digit will be calculate as 828/4 remainder will be 0 so unit digit 2

All these numbers can be verified to not be perfect squares. Option (d) is correct.

It is obvious that the LCM of 5,8,12 and 20 would never be a multiple of 9.
At the same time it has to be a multiple of each of 3, 8 and 5. Option (b) is correct.

x² – 4 = (x – 2) (x + 2) and x² + x – 6 = (x + 3)(x – 2) GCD or HCF of these expressions = (x – 2).
Option (b) is correct.

9¹⁰⁰/8 = (8 + 1)¹⁰⁰/8 → Since this is of the form (a + 1)ⁿ/a, the Remainder = 1. Option (a) is correct.

The condition for the product to be the greatest is if the two terms are equal. Thus, the break up in option (a) would give us the highest product of the two parts.
Option (a) is correct.

Checking each of the options it can be seen that the value in option (c)[viz: 1362480]is divisible by 24.

When the birds sat one on a branch, there was one extra bird. When they sat 2 to a branch one branch was extra.
To find the number of branches, go through options. Checking option (a), if there were 3 branches, there would be 4 birds. (this would leave one bird without branch as per the question.)
When 4 birds would sit 2 to a branch there would be 1 branch free (as per the question).
Hence, the answer (a) is correct.

The number would either be (3n + 1)2 or (3n + 2)2. In the expansion of each of these the only term which would not be divisible by 3 would be the square of 1 and 2 respectively.
When divided by 3, both of these give 1 as remainder. 

For the sum of squares of digits to be 13, it is obvious that the digits should be 2 and 3.
So the number can only be 23 or 32.
Further, the number being referred to has to be 32 since the reduction of 9, reverses the digits.

trying the value in the options you get that the product of 54 × 45 = 2430. Option (a) is correct.

The pairs given in option (d) 78 and 13 and 26 and 39 meet both the conditions of LCM of 78 and HCF of 13.
Option (d) is correct.

The last 3 digits of the number would determine the remainder when it is divided by 8.
The number upto the 120th digit would be 1234567891011… 646.
646 divided by 8 gives us a remainder of 6.

Given:

A candidate attempts 20 questions and scores 72 marks.

The marks awarded for the correct answer is 5.

Marks deducted for the wrong answer is 2.

Concept Used:

Total Marks = Number of question correct × marks given for correct answer - Number of question wrong × marks deduct for wrong answer

Calculation:

Let the number of question answer correct be x.

The number of question answer wrong = 20 - x

Total marks = 72

⇒ 5 × x - (20 - x) × 2 = 72

⇒ 5x - 40 + 2x = 72

⇒ 7x = 112

⇒ x = 16

∴ The questions were answered correctly by him is 16.

3M + 4G + 5W = 750 (i)
6M + 9G + 10W = 1580 (ii)
Adding the two equations we get:
9M + 13G + 15W = 2330 (iii)
Dividing this expression by 3 we get:
3M + 4.33G + 5W =776.666
(iv) - (i) → 0.33 G = 26.666 Æ G = 80
Now, if we look at the equation (i) and multiply it by 2, we get: 6M + 8G + 10W = 1500.
If we subtract the cost of 4 guavas from this we would get: 6M + 4G + 10W = 1500 – 320 = 1180
Option (b) is correct.

The number would be given by the 2 × (LCM of 2, 3 and 4) + 1 → which is 24 + 1 = 25.
Option (a) is correct.