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What is the remainder obtained when 1010 + 105 – 24 is divided by 36?
- a)5
- b)6
- c)12
- d)16
- e)32
Correct answer is option 'E'. Can you explain this answer?
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Verified Answer
What is the remainder obtained when 1010 + 105 – 24 is divided b...
Given:
- Not applicable
To find: The remainder when 1010 + 105 – 24 is divided by 36
Approach:
- Let the required remainder be r. This means, we will be able to write:
1010 + 105 – 24 = 36k + r, where quotient k is an integer and 0 ≤ r < 36
The above expression is our GOAL expression. We’ll try to simplify the dividend 1010 + 105 – 24 till it is comparable to our GOAL expression, and then, by comparison, we’ll be able to find the value of r.
Working Out:
- 1010 + 105 – 24 = 1005 + 1002*10 – (36 – 12)
- =(36*3 – 8)5 + (36*3 – 8)2*10 – 36 + 12
- Now, from Binomial Theorem, we know that every term in the expansion of (36*3 – 8)5 will be divisible by 36, except the last term, and the last term will be (-8)5
- So, we can write: (36*3 – 8)5 is of the form 36a + (-8)5, where 36a is a catch-all term conveying that all the other terms in this expansion are divisible by 36
- Similarly, every term in the expansion of (36*3 – 8)2 will be divisible by 36 except the last term, and the last term will be (-8)2
- So, we can write: (36*3 – 8)2 = 36b + (-8)2
- Now, from Binomial Theorem, we know that every term in the expansion of (36*3 – 8)5 will be divisible by 36, except the last term, and the last term will be (-8)5
- So, the given expression simplifies to:
- {36a + (-8)5 } + {36b + (-8)2}*10 – 36 + 12
- = (36a + 360b – 36) + (-85 + 640 + 12)
- = (36a + 360b – 36) + (-85+ 652)
- = (36a + 360b – 36 + 648) + (-85+ 4)
- = (36a + 360b – 36 + 36*18) + (-85+ 4)
- The above expression is not comparable to our GOAL expression because the term -85 in it is still unresolved. Do we need to calculate the value of -85 to answer this question? No. We only need to express it in terms of 36. Once again, we’ll use Binomial Theorem to do so:
- -85 = -8(82)2 = -8(64)2 = -8(36*2 – 8)2
- Every term in the expansion of (36*2 – 8)2 will be divisible by 36 except the last term. The last term will be (-8)2 = 64
- So, the expression (36*2 – 8)2 can be written as: 36c + 64
- So, -8(36*2 – 8)2 = -8(36c + 64)
- =-8(36c + 36*2 – 8)
- = (-8*36c – 8*36*2) + 64
- -85 = -8(82)2 = -8(64)2 = -8(36*2 – 8)2
- So, the given expression simplifies to: (36a + 360b – 36 + 72) + {(-8*36c – 8*36*2) + 64} + 4
- = (36a + 360b – 36 + 72 – 8*36c – 8*36*2) + 68
- =(36a + 360b – 36 + 72 – 8*36c – 8*36*2 + 36) + 32
- Now, the above expression is exactly comparable to our GOAL Expression: 36k + r
- So, by comparison, we can say that Remainder r = 32
Looking at the answer choices, we see that the correct answer is Option E
Most Upvoted Answer
What is the remainder obtained when 1010 + 105 – 24 is divided b...
To find the remainder when 1010, 105, and 24 are divided by 36, we can perform the division and observe the remainder.
Dividing 1010 by 36:
When we divide 1010 by 36, we get a quotient of 28 and a remainder of 22.
Dividing 105 by 36:
When we divide 105 by 36, we get a quotient of 2 and a remainder of 33.
Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.
Now, let's perform the division again but with the remainders.
Dividing 22 by 36:
When we divide 22 by 36, we get a quotient of 0 and a remainder of 22.
Dividing 33 by 36:
When we divide 33 by 36, we get a quotient of 0 and a remainder of 33.
Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.
Summing the remainders:
To find the remainder when the sum of the three numbers is divided by 36, we sum the remainders obtained in each division: 22 + 33 + 24 = 79.
Reducing the remainder:
Since the remainder obtained (79) is greater than the divisor (36), we need to reduce it. We can do this by repeatedly subtracting the divisor until we obtain a remainder less than the divisor.
79 - 36 = 43
43 - 36 = 7
The remainder after reducing is 7.
Therefore, the remainder obtained when 1010, 105, and 24 are divided by 36 is 7.
Hence, the correct answer is option E.
Dividing 1010 by 36:
When we divide 1010 by 36, we get a quotient of 28 and a remainder of 22.
Dividing 105 by 36:
When we divide 105 by 36, we get a quotient of 2 and a remainder of 33.
Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.
Now, let's perform the division again but with the remainders.
Dividing 22 by 36:
When we divide 22 by 36, we get a quotient of 0 and a remainder of 22.
Dividing 33 by 36:
When we divide 33 by 36, we get a quotient of 0 and a remainder of 33.
Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.
Summing the remainders:
To find the remainder when the sum of the three numbers is divided by 36, we sum the remainders obtained in each division: 22 + 33 + 24 = 79.
Reducing the remainder:
Since the remainder obtained (79) is greater than the divisor (36), we need to reduce it. We can do this by repeatedly subtracting the divisor until we obtain a remainder less than the divisor.
79 - 36 = 43
43 - 36 = 7
The remainder after reducing is 7.
Therefore, the remainder obtained when 1010, 105, and 24 are divided by 36 is 7.
Hence, the correct answer is option E.
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What is the remainder obtained when 1010 + 105 – 24 is divided by 36?a)5b)6c)12d)16e)32Correct answer is option 'E'. Can you explain this answer?
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