"What Is the Least Integer That Is the Sum of Three Different Primes..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

What is the least integer that is the sum of three different primes each greater than 20?

(A) 69

(B) 73

(C) 75

(D) 79

(E) 83

Try it before reading on.

Step 1: Inventory the Constraints

Before computing anything, list every constraint the question places on the answer:

1. The answer must be an integer
2. It must be the sum of three primes
3. The three primes must be different from each other
4. Each prime must be greater than 20
5. The sum must be the least possible (smallest)

Writing these out takes about five seconds and prevents the most common error on this problem: missing the word "different."

Step 2: Identify the Three Smallest Primes Greater Than 20

A prime number has exactly two distinct factors: 1 and itself. By this definition, 1 is not prime (it has only one factor), and 2 is the smallest and only even prime.

Starting from 21 and working upward:

• 21: not prime (divisible by 3 and 7)
• 22: not prime (even, greater than 2)
• 23: prime (first prime greater than 20)
• 24: not prime (even)
• 25: not prime ($5 \times 5$)
• 26: not prime (even)
• 27: not prime ($3 \times 9$)
• 28: not prime (even)
• 29: prime (second prime greater than 20)
• 30: not prime (even)
• 31: prime (third prime greater than 20)

So the three smallest primes greater than 20 are 23, 29, and 31.

If you've memorized your primes up to 100, this step takes about two seconds. If not, testing divisibility by small primes (2, 3, 5, 7) at each number gets you there in under 30 seconds. Knowing your times tables up to $20 \times 20$ makes the non-prime identification much faster, since you can spot composite numbers like $25 = 5 \times 5$ and $27 = 3 \times 9$ without computation.

Step 3: Add the Three Primes

$23 + 29 + 31 = 83$

The answer is (E).

Why This Problem Matters

About 14% of test takers miss this problem, and the most popular wrong answer is (A) 69.

Here's why: the problem says "three different primes," but a meaningful number of people read "three primes" and miss the word "different." They correctly identify 23 as the first prime greater than 20, then add $23 + 23 + 23 = 69$. The math is right for what they thought the question asked. The answer is wrong because the question requires three different primes.

Answer choice (A) exists specifically to catch this error. It's not a random number. It's the exact result of the most common misreading.

A smaller group, about 6%, goes for (B) 73. This likely comes from incorrectly identifying 21 as prime ($21 + 23 + 29 = 73$) or from including 19, which is prime but not greater than 20 ($19 + 23 + 31 = 73$). Both errors trace back to either a content gap (not knowing 21 isn't prime) or a constraint miss (not catching "greater than 20").

The fix for both error types is the constraint inventory. If you've written "different" and "greater than 20" on your list before computing, neither trap catches you. And if you do a quick re-read of the question before committing to your answer, you'll catch the word "different" even if you missed it the first time.

The deeper takeaway: invest in knowing your primes up to 100 if you're aiming for 655 or higher. The front-loaded memorization pays off in speed and confidence under pressure. If your target score is lower, the divisibility-testing approach works fine, but practice it enough that you can execute it without hesitation.

Want the full strategy behind this problem? Read: GMAT® Quant Fundamentals: Process Over Computation

From Episode 33 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "If (4 − x)/(2 + x) = x, What Is the Value of x² + 3x − 4?" — GMAT® Worked Solution.