Number System
- The sum of all natural numbers between 100 and 200, which are multiples of 3 is :
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Numbers divisible by 3 and lying between 100 and 200 are : 102, 105, 108, 111,...................... 198
Let number of terms = n
As we know the formula
an = a + ( n – 1 )d
where an = Nth Term a = First term , n = Number of terms and d = Common Difference
∴ 198 = 102 + ( n – 1 ) 3⇒ n − 1 = 198 − 102 = 32 3
⇒ n = 33∴ Sn = Number of terms = ( First term + Last term ) 2
Correct Option: B
Numbers divisible by 3 and lying between 100 and 200 are : 102, 105, 108, 111,...................... 198
Let number of terms = n
As we know the formula
an = a + ( n – 1 )d
where an = Nth Term a = First term , n = Number of terms and d = Common Difference
∴ 198 = 102 + ( n – 1 ) 3⇒ n − 1 = 198 − 102 = 32 3
⇒ n = 33∴ Sn = Number of terms = ( First term + Last term ) 2 ∴ Sn = n = ( a + l ) 2 ∴ Sn = 32 (102 + 198) = 4950 2
- The least number of five digits which has 123 as a factor is
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The smallest number of 5 digits = 10000
Remainder on dividing 10000 by 123 = 37Correct Option: B
The smallest number of 5 digits = 10000
Remainder on dividing 10000 by 123 = 37
∴ Required number
= 10000 + (123 – 37) = 10086
- The sum of the four consecutive even numbers is 284. What would be the smallest number.
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Let four consecutive even numbers are P, P + 2, P + 4 and P + 6
According to the question,
P + P + 2 + P + 4 + P + 6 = 284Correct Option: C
Let four consecutive even numbers are P, P + 2, P + 4 and P + 6
According to the question,
P + P + 2 + P + 4 + P + 6 = 284
⇒ 4P + 12 = 284
⇒ 4P = 284 - 12 = 272
∴ P = 272/4 = 68
- The sum of the digits of a two-digit number is 14 and the difference between the two digits of the number is 2. What is the product of the two digits of the two-digit number ?
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Let be the ten's digit be P and unit's digit be Q.
The two-digit number = 10P + Q
(where, P > Q)
According to the question,
P + Q = 14 ......(i)
and P - Q = 2 .......(ii)Correct Option: B
Let be the ten's digit be P and unit's digit be Q.
The two-digit number = 10P + Q
(where, P > Q)
According to the question,
P + Q = 14 ......(i)
and P - Q = 2 .......(ii)
solving Eqs. (i) and (ii), we get
P = 8 and Q = 6
∴ Required product = 8 x 6 = 48
- The unit digit in the sum of (124)372 + (124)373 is
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As we know that
41 = 4;
42 = 16;
43 = 64;
44 = 256;
45 = 1024;
As we see that
Unit place of 41 = 4;
Unit place of 42 = 16;
Unit place of 43 = 64;
Unit place of 44 = 256;
Unit place of 45 = 1024;
Unit digit is repeating after every 4th index number.
Remainder on dividing 372 by 4 = 0
Remainder on dividing 373 by 4 = 1Correct Option: D
As we know that
41 = 4;
42 = 16;
43 = 64;
44 = 256;
45 = 1024;
As we see that
Unit place of 41 = 4;
Unit place of 42 = 16;
Unit place of 43 = 64;
Unit place of 44 = 256;
Unit place of 45 = 1024;
Unit digit is repeating after every 4th index number.
Remainder on dividing 372 by 4 = 0
Remainder on dividing 373 by 4 = 1
∴ Required unit digit = Unit digit of (124)372 + (124)373
∴ Required unit digit = Unit digit of (4)372 + (4)373
as we know that Remainder on dividing 372 by 4 = 0 and Remainder on dividing 373 by 4 = 1
∴ Required unit digit = Unit digit of (4)4 + (4)1
∴ Required unit digit = Unit digit of of sum of ( 6 + 4 )
∴ Required unit digit = Unit digit of of sum of ( 10 )
∴ Required unit digit = 0
