Test Level 1: Number System - 1 - CAT MCQ

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.

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 × 3⁴. Hence: 15 × 27
45 × 9 are the only two options. 
From these factors pairs only the second pair gives us the desired result. 
i.e. Number × sum of digits = 405.
Hence, the answer is 45.

Let the numbers be x and y.
So according to question,
x - y = 48

or (√x - √y)(√x + √y) = 48......eqn .1

arithmetic mean of x and y = (x + y)/2

and geometric mean of x and y = √(xy)

so according to question,
(x + y)/2 - √(xy) = 18

so x + y - 2√xy = 36
(√x - √y)² = 36
√x - √y = ±6.....eqn. 2

Putting eqn. 2 in eqn. 1
we get,
√x + √y = ±8....eqn.3

Adding eqn.2 and eqn.3,
√x + √y + √x - √y =14
2√x = 14
√x =7

So x = 7² = 49....eqn.4

As x - y = 48
y = x - 48 = 49 - 48 = 1.

Therefore x = 49 and y = 1

For 381A 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.

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

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.

For the number 15B to be divisible by 6, it must satisfy two conditions:

1. Divisibility by 2: The number must be even.
2. Divisibility by 3: The sum of its digits must be divisible by 3.

Since 15B represents a number with digits 1, 5, and B, let’s analyze each option for B:

• For divisibility by 2, B must be even (i.e., 0, 2, 4, 6, or 8) because an even B makes 15B an even number.
• For divisibility by 3, the sum 1+5+B=6+B must be divisible by 3.

Since 6 is already divisible by 3, any even B (0, 2, 4, 6, or 8) will keep the sum divisible by 3.

The correct answer is: (d) Both (a) and (c)

Also, value of B is divisible by 6. Therefore, both options are correct, B will be even and B will be divisible by 6.

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 3⁵⁴ will give a units digit equivalent to 3⁴ⁿ⁺² which would give us a unit digit of 3² i.e.9).

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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.