Which of the following are multiples?

To determine which Roman numerals are meaningless, we need to apply the rules for forming Roman numerals:

• Only I, X, C, M can be repeated, up to three times. V, L, D cannot be repeated.
• A smaller numeral can only precede a larger numeral if it is I (before V or X), X (before L or C), or C (before D or M).
• V, L, D cannot be subtracted from any numeral.

Let's examine each option:

a) XLIV

• XL means $50 - 10 = 40$.
• IV means $5 - 1 = 4$.
• XLIV means $40 + 4 = 44$. This is a valid Roman numeral.

b) VL

• V means 5, L means 50.
• This attempts to represent $50 - 5 = 45$.
• However, V cannot be subtracted from L. Only I can precede V or X for subtraction. This is a meaningless Roman numeral.

c) XXI

• XX means $10 + 10 = 20$.
• I means 1.
• XXI means $20 + 1 = 21$. This is a valid Roman numeral.

d) XCI

• XC means $100 - 10 = 90$.
• I means 1.
• XCI means $90 + 1 = 91$. This is a valid Roman numeral.

e) IC

• I means 1, C means 100.
• This attempts to represent $100 - 1 = 99$.
• However, I can only be subtracted from V or X. It cannot be subtracted from C. The correct way to write 99 is XCIX. This is a meaningless Roman numeral.

f) LII

• L means 50.
• II means $1 + 1 = 2$.
• LII means $50 + 2 = 52$. This is a valid Roman numeral.

g) LIV

• L means 50.
• IV means $5 - 1 = 4$.
• LIV means $50 + 4 = 54$. This is a valid Roman numeral.

h) XIX

• X means 10.
• IX means $10 - 1 = 9$.
• XIX means $10 + 9 = 19$. This is a valid Roman numeral.

i) III

• III means $1 + 1 + 1 = 3$. This is a valid Roman numeral.

j) VII

• V means 5.
• II means $1 + 1 = 2$.
• VII means $5 + 2 = 7$. This is a valid Roman numeral.

The meaningless Roman numerals are:

• b) VL
• e) IC