Remainder Theorem

Find the remainder of a large numbers divided by a numbers:

Step 1: The remainder of multiplication of a numbers by a value is same as multiplication of remainder of individual terms.

1421 * 1423 * 1425 / 12 same as R (1421/12) * R(1423/12) * R(1425/12) where R(p/q) is remainder of p/q. Therefore we get 5 * 7 * 9 / 12 = 35 * 9 / 12 = 99 / 12 = 3.

Method II — Find the remainder of a large numbers divided by a numbers:

Step 1: The remainder of multiplication of a numbers by a value is same as multiplication of remainder of individual terms.

11 * 10 * 9 / 12 same as NR (11/12) * NR(10/12) * NR(9/12) where NR(p/q) is negative remainder of p/q.

So if 35/12 then remainder shall be 11 and negative remainder shall 11 – 12 = -1

Step 2: -1 * -2 * -3 / 12 = – 6 so remainder is 6.

Calculate remainder when dealing with large powers:

Step 1: Can question be converted to following formulae.(ax + 1)n / a = Remainder is 1 and (ax – 1)n / a = Remainder is -1 i.e. a-1.

E.g: 37124556 / 9 = (9*4 + 1)124556 / 9 = Remainder is 1

35124556 / 9 = (9*4 – 1)124556 / 9 = Remainder is -1 i.e 9-1 = 8

Find the last two digits of a large multiplication:

Step 1: The last two digits of multiplication of a numbers by a value is same as finding the remainder of multiplication terms when divided by 100.

1421 * 1423 * 1425 / 100 same as R (1421/100) * R(1423/100) * R(1425/100) where R(p/q) is remainder of p/q.

Therefore we get 21 * 23 * 25 / 100 = 525 * 23 / 100 ; 25 * 23 / 100 = 575 / 100

Thus the last two digits are 75.

Find the last digits of a large multiplication:

Step 1: The last digit of multiplication of a numbers by a value is same as finding the remainder of multiplication terms when divided by 10.

1421 * 1423 * 1425 / 10 same as R (1421/10) * R(1423/10) * R(1425/10) where R(p/q) is remainder of p/q.

Therefore we get 1 * 3 * 5 / 10 = 15 / 10 = 5

Thus the last digit is 5.

Remainder Theorem - Dealing with large powers

If you can express the expression in the form (ax+1)na the remainder will become 1 directly. In such a case, no matter how large the value of the power n is, the remainder is 1.

• Example : (3712654)9=(9∗4+1)126549=1
• If you can express the expression in the form (ax−1)na the remainder will become 1 directly. In such a case, no matter how large the value of the power n is, the remainder is 1.
• Example : (3512654)9=(9∗4−1)126549=−1. The remainder cannot be negative and so -1 remainder means 8.
• If you can express the expression in the form (a)na+1 the remainder will become “a” directly if “n” is odd and 1 if “n” is even.
• If you can express the expression in the form (a+1)na the remainder will become 1 for any power “n”.

Solved Question Papers

Q. Find the remainder when 73 + 75 + 78 + 57 + 197 is divided by 34

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Q. Find the remainder when 73 * 75 * 78 * 57 * 197 is divided by 34.

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Q. Find the remainder when 73 * 75 * 78 * 57 * 197 * 37 is divided by 34

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Q. Find the remainder when 43197 is divided by 7

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Q. Find the remainder when 51203 is divided by 7

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Q. Find the remainder when 5928 is divided by 7.

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