Find the one which does not belong to that group ?
Answer: B
All four are perfect squares, so look at what is being squared.
\[ 36 = 6^{2}, \quad 49 = 7^{2}, \quad 64 = 8^{2}, \quad 81 = 9^{2} \]
The roots 6, 8 and 9 are composite; 7 is prime.
So 49 does not belong - the square of the only prime root.
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Find the one which does not belong to that group ?
Answer: C
All four are perfect cubes, so look at the roots.
\[ 8 = 2^{3}, \quad 27 = 3^{3}, \quad 64 = 4^{3}, \quad 125 = 5^{3} \]
The roots 2, 3 and 5 are prime; 4 is composite.
So 64 does not belong.
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Find the one which does not belong to that group ?
Answer: B
Three of the four are perfect cubes.
\[ 343 = 7^{3}, \quad 1331 = 11^{3}, \quad 2197 = 13^{3} \]
\[ 121 = 11^{2} \text{, a square rather than a cube} \]
So 121 does not belong. It is deliberately paired with 1331, the cube of the same number 11.
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Find the one which does not belong to that group ?
Answer: A
Three of the four are multiples of 3.
\[ 48 = 3 \times 16, \quad 75 = 3 \times 25, \quad 84 = 3 \times 28 \]
\[ 35 = 5 \times 7 \]
So 35 does not belong. Its digit sum is 8, which is not divisible by 3, while the others give 12, 12 and 12.
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Find the one which does not belong to that group ?
Answer: C
Three of the four numbers read the same forwards and backwards.
\[ 42624 \to 42624 \quad\checkmark \qquad 37573 \to 37573 \quad\checkmark \]
\[ 93339 \to 93339 \quad\checkmark \qquad 84284 \to 48248 \quad\times \]
So 84284 is not a palindrome and does not belong. Checking only the outer digits is not enough - 84284 starts and ends with different digits, which settles it at a glance.
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Find the one which does not belong to that group ?
Answer: A
Three of the four are multiples of 9.
\[ 27 = 9 \times 3, \quad 36 = 9 \times 4, \quad 45 = 9 \times 5 \]
\[ 30 = 9 \times 3 + 3 \]
So 30 does not belong. Digit sums make it immediate: 9, 9 and 9 against 3.
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Find the one which does not belong to that group ?
Answer: C
All four use two 4s and two 2s, so look at how they are arranged.
In 4422, 2442 and 2244 the like digits sit together as pairs.
In 4242 they alternate, so no two identical digits are adjacent.
So 4242 does not belong.
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Find the one which does not belong to that group ?
Answer: C
Three of the four are products of two consecutive integers.
\[ 20 = 4 \times 5, \quad 42 = 6 \times 7, \quad 72 = 8 \times 9 \]
\[ 58 = 2 \times 29 \text{, and } 7 \times 8 = 56, \; 8 \times 9 = 72 \]
So 58 does not belong - it falls between two such products without being one.
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Find the one which does not belong to that group ?
Answer: B
Three of the four are not divisible by 9.
\[ 630 = 9 \times 70 \quad (\text{digit sum } 6+3+0 = 9) \]
\[ 30 \to 3, \qquad 10 \to 1, \qquad 520 \to 7 \]
Only 630 has a digit sum divisible by 9, so 630 does not belong.
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Find the one which does not belong to that group ?
Answer: C
Compare the digit sums.
\[ 508 \to 5+0+8 = 13, \qquad 328 \to 3+2+8 = 13, \qquad 148 \to 1+4+8 = 13 \]
\[ 608 \to 6+0+8 = 14 \]
So 608 does not belong - the only one whose digits do not add to 13.
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