ComputePrep Timed quant & vedic drills

Nikhilam

Nikhilam multiplication — two numbers near a base, in two steps

Nikhilam multiplication multiplies two numbers close to a base such as 100. Write each number's deficit from the base, subtract one number's deficit from the other number crosswise to get the front, and multiply the two deficits to get the back. For 97 × 96: 97 − 4 = 93 and 3 × 4 = 12, giving 9312.

Nikhilam is the most dramatic of the Vedic shortcuts when it applies — a two-digit multiplication that would take real effort becomes two single-digit operations. Its limitation is equally clear: both numbers must sit near the same base, so it is an opportunistic technique rather than a general one.

Try it against the clock. Nikhilam runs at 20s easy, 25s medium and 35s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.

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How nikhilam multiplication works, step by step

  1. Write each number's deficit from the base

    For 97 × 96 with base 100, the deficits are 3 and 4. Write them beside each number with a minus sign. Getting the base right matters — both numbers must be near the same one.

  2. Subtract crosswise for the front half

    Take either number minus the OTHER's deficit: 97 − 4 = 93, or equivalently 96 − 3 = 93. That both routes agree is a built-in check that costs nothing to use.

  3. Multiply the deficits for the back half

    3 × 4 = 12. Pad to the base's digit count — two digits for base 100 — so the answer is 9312. This padding step is where most errors occur.

  4. Handle numbers above the base with surpluses

    For 103 × 104 the surpluses are 3 and 4, so add crosswise (103 + 4 = 107) and multiply the surpluses (12), giving 10712. Same method, opposite sign.

Choosing the base and reading the result

SituationWhat to do
Both just below 100Subtract crosswise, multiply deficits
Both just above 100Add crosswise, multiply surpluses
One above, one belowCrosswise still works; the back part goes negative
Both near 1000Back part takes three digits
Back part overflows its widthCarry the excess into the front
Numbers not near any baseUse crosswise multiplication instead

3 real nikhilam multiplication, with worked solutions

Generated by the same engine that mints the ComputePrep daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.

Question 1

111 × 111 = ?

Nikhilam multiplication — beat the clock.

  1. 12321
  2. 12324
  3. 11211
  4. 22200
  5. 122121
Show the worked solution
  1. Choose the base (nearest power of 10): 100.
  2. Distance from base: 111 is 11 above 100, 111 is 11 above 100.
  3. Cross-add: 111 + 11 = 122 (same as 111 + 11).
  4. Multiply the distances: 11 × 11 = 121.
  5. Base 100 has 2 zeros, so the right-hand part must occupy exactly 2 digits.
  6. 121 needs 3 digits — one too many, so carry 1 into the left part.
  7. Left part: 122 + 1 = 123. Right part: 21.
  8. Write them side by side: 123 | 21 = 12321.

Question 2

1005 × 1013 = ?

Nikhilam multiplication — beat the clock.

  1. 101865
  2. 1018071
  3. 1018065
  4. 1006013
  5. 2018000
Show the worked solution
  1. Choose the base (nearest power of 10): 1000.
  2. Distance from base: 1005 is 5 above 1000, 1013 is 13 above 1000.
  3. Cross-add: 1005 + 13 = 1018 (same as 1013 + 5).
  4. Multiply the distances: 5 × 13 = 65.
  5. Base 1000 has 3 zeros, so the right-hand part must occupy exactly 3 digits.
  6. 65 fits, padded to 3 digits → 065.
  7. Left part: 1018. Right part: 065.
  8. Write them side by side: 1018 | 065 = 1018065.

Question 3

85 × 96 = ?

Nikhilam multiplication — beat the clock.

  1. 18100
  2. 8596
  3. 9685
  4. 8171
  5. 8160
Show the worked solution
  1. Choose the base (nearest power of 10): 100.
  2. Distance from base: 85 is 15 below 100, 96 is 4 below 100.
  3. Cross-subtract: 85 − 4 = 81 (same as 96 − 15).
  4. Multiply the distances: 15 × 4 = 60.
  5. Base 100 has 2 zeros, so the right-hand part must occupy exactly 2 digits.
  6. 60 fits, padded to 2 digits → 60.
  7. Left part: 81. Right part: 60.
  8. Write them side by side: 81 | 60 = 8160.

Common mistakes this drill is built from

The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.

Not padding the back half

For base 100 the back is two digits, so 98 × 97 gives 95 and 06, that is 9506 — not 956. Padding is the single most common nikhilam error.

Using different bases for the two numbers

Both numbers must be near the same base. 97 × 48 is not a nikhilam question, and forcing it produces nonsense rather than a wrong-by-a-little answer.

Forgetting the carry when the deficit product is large

If the deficits multiply past the base's width, the excess carries into the front half exactly as in ordinary addition.

FAQ

What is nikhilam multiplication?

A Vedic method for multiplying two numbers close to a base like 100. You subtract crosswise to get the leading part of the answer and multiply the two deficits to get the trailing part. For 97 × 96 that is 93 and 12, giving 9312 — two easy operations instead of a full multiplication.

When does nikhilam not work?

When the numbers are not near a common base. 97 × 96 is ideal; 97 × 48 is not, because 48 is nowhere near 100. In those cases the general crosswise method is the right tool, and recognising which case you are in is part of using shortcuts well.

Can nikhilam handle numbers above the base?

Yes — use surpluses instead of deficits and add crosswise rather than subtracting. For 103 × 104: 103 + 4 = 107 and 3 × 4 = 12, giving 10712. The structure of the method is unchanged.

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