IPMAT Indore 2026 · Quantitative Ability · Question 30

IPMAT Indore 2026 previous year question on Prime Numbers and Co-primes, with the answer and a step-by-step solution.

Prime Numbers and Co-primesNumber SystemMultiple choiceMedium
If a, b, and c are three prime numbers such that abc=23\left( a+b+c \right), then the maximum possible value of a+b-c is
  1. A
    33
  2. B
    46
  3. C
    28
  4. D
    25
Pick an option, then check.
Answer & solution

Answer: Option A

Step 1 — use the fact that 23 is prime. Since 23 divides abc and all three are prime, one of them must be 23. Call the other two p and q.

Step 2 — simplify.

23pq=23\left( p+q+23 \right) \Rightarrow pq=p+q+23

Step 3 — factorise. Subtract p+q and add 1 to both sides:

pq-p-q+1=24 \Rightarrow \left( p-1 \right)\left( q-1 \right)=24

Step 4 — find prime pairs. Check each factorisation of 24:

\left( p-1,\,q-1 \right)

\left( p,\,q \right)

both prime?

\left( 1,24 \right)

\left( 2,25 \right)

✗

\left( 2,12 \right)

\left( 3,13 \right)

✓

\left( 3,8 \right)

\left( 4,9 \right)

✗

\left( 4,6 \right)

\left( 5,7 \right)

✓

So the triple is \{3,13,23\} or \{5,7,23\}.

Step 5 — maximise a+b-c. Make c the smallest member:

\{3,13,23\}: 13+23-3=33

\{5,7,23\}: 7+23-5=25

\max =33

Answer: Option 1 — 33

Shortcut. Two moves win it: 23 must be one of the primes, and pq-p-q=23 begs to be written \left( p-1 \right)\left( q-1 \right)=24. Then remember a+b-c is not symmetric — assign the smallest prime to c to maximise it.

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