RAS Pre 2021, held on 27 October 2021. Question text as printed in RPSC's booklet; answer as per RPSC's final answer key.

Arithmetic mean of the numbers between 1 to 156 which are divisible by 2, 3, 4 and 6 is -

Q: What is the answer to RAS Pre 2021 Q112?

(3). Correct — option (3), 78. Step 1 — replace the four divisors by one. A number divisible by 2, 3, 4 and 6 is divisible by their LCM. LCM (2, 3, 4, 6) = 12 Step 2 — list the multiples of 12 between 1 and 156. Read "between 1 to 156" as leaving out the end numbers, the reading that fits the options; 156 = 12 × 13 is then left out. 12, 24, 36, … , 144 Count = 12 Step 3 — add them. Sum = 12 × (1 + 2 + … + 12) = 12 × 78 = 936 Step 4 — divide by the count. Mean = 936 ÷ 12 = 78 Check: the list is evenly spaced, so mean = (first + last) ÷ 2 = (12 + 144) ÷ 2 = 78. If 156 were counted, the list would run from 12 to 156 and the mean would be (12 + 156) ÷ 2 = 84, which is not among the options. The idea to remember: turn several divisors into their LCM, then average an evenly spaced list as (first + last) ÷ 2.

(1)76
(2)74
(3)78
(4)91

Answer

(3)

Why

Correct — option (3), 78.

Step 1 — replace the four divisors by one.
A number divisible by 2, 3, 4 and 6 is divisible by their LCM.
LCM (2, 3, 4, 6) = 12

Step 2 — list the multiples of 12 between 1 and 156.
Read "between 1 to 156" as leaving out the end numbers, the reading that fits the options; 156 = 12 × 13 is then left out.
12, 24, 36, … , 144
Count = 12

Step 3 — add them.
Sum = 12 × (1 + 2 + … + 12) = 12 × 78 = 936

Step 4 — divide by the count.
Mean = 936 ÷ 12 = 78

Check: the list is evenly spaced, so mean = (first + last) ÷ 2 = (12 + 144) ÷ 2 = 78.

If 156 were counted, the list would run from 12 to 156 and the mean would be (12 + 156) ÷ 2 = 84, which is not among the options.

The idea to remember: turn several divisors into their LCM, then average an evenly spaced list as (first + last) ÷ 2.

Why the others are wrong

(1)
76 — The mean of a run of consecutive multiples of 12, from 12a to 12b, is (12a + 12b) ÷ 2 = 6(a + b). It is therefore always a multiple of 6.
76 ÷ 6 = 12⅔, so 76 cannot be the mean of such a list. Among the four options, only 78 = 6 × 13 passes this test.
(2)
74 — A mean of 74 for an evenly spaced list would need first + last = 2 × 74 = 148.
The multiples of 12 between 1 and 156 start at 12 and end at 144, which add up to 156, not 148. 74 is also not a multiple of 6 (74 ÷ 6 = 12⅓).
(4)
91 — 91 lies above the middle of the list. The twelve multiples pair off from the two ends: 12 + 144, 24 + 132, … , 72 + 84, each pair adding to 156.
Six pairs of 156 give 936, and 936 ÷ 12 = 78. A mean of 91 would need a total of 91 × 12 = 1,092.

Concept

The arithmetic mean of a set of values is their sum divided by how many there are.

A number is divisible by each of several numbers exactly when it is divisible by their least common multiple (LCM). The LCM of 2, 3, 4 and 6 is 12, not their product 144.

Multiples of a number form an evenly spaced list, an arithmetic progression. Its sum is (number of terms) × (first + last) ÷ 2, so its mean is simply (first + last) ÷ 2.

In an evenly spaced list the mean equals the median. Here the two middle terms are 72 and 84, and (72 + 84) ÷ 2 = 78.

RPSC's 2021 syllabus lists "Mean(Arithmetic, Geometric and Harmonic), Median and Mode" under Basic Numeracy in Reasoning & Mental Ability.

The arithmetic mean is the familiar average. The geometric mean of n positive values is the nth root of their product, and the harmonic mean is n divided by the sum of their reciprocals.

For positive values that are not all equal, the arithmetic mean is greater than the geometric mean, which is greater than the harmonic mean.

Key facts

A number divisible by each of 2, 3, 4 and 6 is divisible by their LCM, 12.
Multiples of 12 between 1 and 156, leaving out 156: 12, 24, … , 144, twelve numbers adding up to 936.
Arithmetic mean = sum of values ÷ number of values; here 936 ÷ 12 = 78.
For an evenly spaced list, mean = (first term + last term) ÷ 2.
Counting 156 as well would give thirteen multiples with mean (12 + 156) ÷ 2 = 84, which is not among the options.

From four divisors to the mean

LCM (2, 3, 4, 6)→12

Multiples of 12 between 1 and 156→12, 24, … , 144 (12 numbers)

Sum→12 × 78 = 936

Mean = 936 ÷ 12→78 — option (3)

Check: (12 + 144) ÷ 2→78

156 is left out, the reading that fits the options; counting it would give a mean of 84, which is not an option.

Study next

LCM and HCF of a set of numbersSum of an arithmetic progressionMedian and mode of a listGeometric and harmonic meansCorrecting a mean after a wrong entry

Common traps

Using the product of the divisors. 2 × 3 × 4 × 6 = 144 is a common multiple, but the LCM is 12; multiples of 144 would leave out 12, 24 and the rest.
Overlooking how "between" is read. With 156 included the list has thirteen terms and a mean of 84, which is not an option; leaving 156 out gives 78.
Averaging 1 and 156. The list starts at 12, the first multiple of 12, not at 1.

A question can ask for the mean of the numbers in a range that are divisible by given numbers, as this one does.

A question can also ask for the sum or count of such numbers, or for the mean after one value is added, removed or corrected.

Related PYQs

RPSC_2024_PRE_GS_Q962024Open & attempt →

The mean of 100 observations was calculated as 49. It was later discovered that three observations taken as 40, 20, 50 were actually 60, 70, 80 respectively. The correct mean is

(1) 48
(2) 49.33
(3) 50
(4) 49.5

Answer(3)

Same arithmetic mean as sum ÷ count. That question corrects the mean of 100 observations after three values were wrongly taken (RPSC's key: 50); this one first builds the list, the multiples of 12 between 1 and 156, and then averages it.

Practice

practice — not a real PYQ
The arithmetic mean of all numbers between 1 and 100 that are divisible by both 4 and 6 is –
- (a)48
- (b)54
- (c)50
- (d)60
Answer(2) — LCM (4, 6) = 12, so the numbers are 12, 24, … , 96 and the mean is (12 + 96) ÷ 2 = 54.
Option (1) is the mean of 12 to 84, which leaves out 96; option (3) is the mean of every whole number from 1 to 99; option (4) is the mean of the multiples of 24 (4 × 6), which skips 12, 36, 60 and 84.
practice — not a real PYQ
The mean of five numbers is 18. When one of the numbers is removed, the mean of the remaining four is 16. The number removed is –
- (a)2
- (b)26
- (c)18
- (d)17
Answer(2) — Sum of five = 5 × 18 = 90 and sum of four = 4 × 16 = 64, so the removed number is 90 − 64 = 26. Option (1) is the fall in the mean, option (3) is the old mean, and option (4) is the average of the two means.