number system Model Questions & Answers, Practice Test for ssc cgl tier 2
N! is completely divisible by $13^52$. What is sum of the digits of the smallest such number N?
Answer: (a)
The number needs to be less than 13 × 52 = 676.
The highest power of 13 in 676! is 56.
The power of 13 in the smallest such number needs to be exactly 52.
If we subtract 13 × 3 = 39 from 676, we get 637.
The number 637! will be the smallest number of type N! that is completely divisible by 1352.
The sum of the digits of 637 is 16.
The largest integer that divides product of any four consecutive integers is
Answer: (d)
The largest integer that divides product of any four consecutive integers is 4!
i.e., 24. e.g., 1, 2, 3, 4 are four consecutive integers.
Multiplication = 1 × 2 × 3 × 4 = 24 which is divided by 24.
Consider the following statements in respect of three 3- digit numbers XYZ, YZX and ZXY :
- The sum of the numbers is not divisible by (X + Y + Z).
- The sum of the numbers is divisible by 111.
Answer: (a)
Statement 1
XYZ | = | 100X | + | 10Y | + | Z |
YZX | = | 100Y | + | 10Z | + | Z |
ZXY | = | 100Z | + | 10X | + | Y |
= 111 (X + Y + Z), which is divisible by (X + Y + Z)
So, it is not correct
Statement 2
XYZ | = | 100X | + | 10Y | + | Z |
YZX | = | 100Y | + | 10Z | + | X |
ZXY | = | 100Z | + | 10X | + | Y |
= 111 (X + Y + Z), which is divisible by 111
So, it is correct
The digit in the unit's place of the number represented by $(7^95 –3^58)$ is :
Answer: (c)
Unit digit in $7^4$ is 1.
So, unit digit in $7^92$ is 1.
∴ Unit digit in $7^95$ is 3.
Unit digit in $3^4$ is 1.
∴ Unit digit in $3^56$ is 1.
∴ Unit digit in $3^58$ is 9.
∴ Unit digit in $(7^95 – 3^58)$
= (13 – 9) = 4.
Let a two-digit number be k times the sum of its digits. If the number formed by interchanging the digits is m times the sum of the digits, then the value of m is
Answer: (b)
Let two digit number = 10y + x
According to question,
Ist condition, 10y + x = k(x + y) ...(1)
2nd condition, 10x + y = m(x + y) ...(2)
Adding (1) and (2) we get
11x + 11y = (k + m) × (x + y)
11(x + y) = (k + m) (x + y)
k + m = 11 ⇒ m = 11 – k
∴ Option (c) is correct
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