Number System
- The numbers 2272 and 875 are divided by a 3-digit number N, giving the same remainders. The sum of the digits of N is
-
View Hint View Answer Discuss in Forum
Let the remainder in each case be x.
Then, (2272 – x) and (875 – x) are exactly divisible by that three digit number.
Hence, their difference [(2272 – x) – (875 – x)] = 1397 will also be exactly divisible by the said divisor (N).
Now, 1397 = 11×127 Since both 11 and 127 are prime numbers, N is 127.
∴ Sum of digits = 1+ 2 + 7 = 10Correct Option: A
Let the remainder in each case be x.
Then, (2272 – x) and (875 – x) are exactly divisible by that three digit number.
Hence, their difference [(2272 – x) – (875 – x)] = 1397 will also be exactly divisible by the said divisor (N).
Now, 1397 = 11×127 Since both 11 and 127 are prime numbers, N is 127.
∴ Sum of digits = 1+ 2 + 7 = 10
- A 2-digit number is 3 times the sum of its digits. If 45 is added to the number, its digits are interchanged. The sum of digits of the number is
-
View Hint View Answer Discuss in Forum
Let the two digit number be = 10x + y.
According to the question,
10 x + y = 3(x + y)
⇒ 10 x + y = 3x + 3y
⇒ 10 x + y – 3x –3y = 0
⇒ 7 x – 2y = 0 ....(i)
and,
10 x + y + 45= 10 y + x
⇒ 10 y + x – 10x –y = 45
⇒ 9 y – 9 x = 45
⇒ 9 (y – x) = 45
⇒ y – x = 5 ...(ii)
2 × (ii) + (i) we have
2y –2x + 7x – 2y =10⇒ 5x = 10 ⇒ x = 10 = 2 5
From equation (ii),
y –2 = 5 ⇒ y = 2 + 5 = 7
∴ Number = 10x + y
= 2 × 10 +7 = 27
∴ Sum of digits = 2 + 7 = 9Correct Option: B
Let the two digit number be = 10x + y.
According to the question,
10 x + y = 3(x + y)
⇒ 10 x + y = 3x + 3y
⇒ 10 x + y – 3x –3y = 0
⇒ 7 x – 2y = 0 ....(i)
and,
10 x + y + 45= 10 y + x
⇒ 10 y + x – 10x –y = 45
⇒ 9 y – 9 x = 45
⇒ 9 (y – x) = 45
⇒ y – x = 5 ...(ii)
2 × (ii) + (i) we have
2y –2x + 7x – 2y =10⇒ 5x = 10 ⇒ x = 10 = 2 5
From equation (ii),
y –2 = 5 ⇒ y = 2 + 5 = 7
∴ Number = 10x + y
= 2 × 10 +7 = 27
∴ Sum of digits = 2 + 7 = 9
- On multiplying a number by 7, all the digits in the product appear as 3’s. the smallest such number is
-
View Hint View Answer Discuss in Forum
Let the smallest number be x.
∴ x × 7 = 33333....⇒ x = 33333.... = 47619 7
Correct Option: C
Let the smallest number be x.
∴ x × 7 = 33333....⇒ x = 33333.... = 47619 7
- 7 is added to a certain number; the sum is multiplied by 5; the product is divided by 9 and 3 is subtracted from the quotient. Thus if the remainder left is 12, what was the original number ?
-
View Hint View Answer Discuss in Forum
Check through options
20 → 20 + 7 = 27 → 27 × 5
= 135 → 135 ÷ 9
= 15 → 15 – 3 = 12
OR, We will solve the problem from the opposite side.
Here the remainder is 12.
12 + 3 = 15
15 × 9 = 135
135 ÷ 5 = 27
27 – 7 = 20
∴ The original number was 20.Correct Option: B
Check through options
20 → 20 + 7 = 27 → 27 × 5
= 135 → 135 ÷ 9
= 15 → 15 – 3 = 12
OR, We will solve the problem from the opposite side.
Here the remainder is 12.
12 + 3 = 15
15 × 9 = 135
135 ÷ 5 = 27
27 – 7 = 20
∴ The original number was 20.
- If the difference of two numbers is 3 and the difference of their squares is 39, then the larger number is
-
View Hint View Answer Discuss in Forum
Let the numbers be a and b,
where a > b
According to the question,
a – b = 3 ....(i)
a2 − b2 = 39
⇒ (a + b) (a – b) = 39⇒ a + b = 39 = 39 = 13 a − b 3
⇒ a + b = 3 .....(ii)
Adding equations (i) and (ii)
2a = 16 ⇒ a = 8
Correct Option: A
Let the numbers be a and b,
where a > b
According to the question,
a – b = 3 ....(i)
a2 − b2 = 39
⇒ (a + b) (a – b) = 39⇒ a + b = 39 = 39 = 13 a − b 3
⇒ a + b = 3 .....(ii)
Adding equations (i) and (ii)
2a = 16 ⇒ a = 8
